I ?\.v. m n RitoiiiiiiliiWitaiiii'iliiiiiii'l'iii IMdftBMKMtMmwit'lCJ'WilJftJvAiitKMll^^ CORNELL UNIVERSITY LIBRARY MATHEMATICS Cornell University Library QA 196.T13 1873 An elementary treatise on quaternions 3 1924 001 570 971 Moil tii QUATERNIONS TAIT SonDon : Cambridge warehouse, 17, PATERNOSTER ROW. ©amfiriDgc: deighton, bell, and co. AN ELEMENTAEY TREATISE ON QUATEKNIONS BY P. G. TAIT, M.A. FORMEKLY FELLOW OF SI. PETER'S COLLEGE, CAMBRIDGE PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH TfKpaKTVV, nayav aevaov <f)V(rfas pif^iijiar txovaav. SECOND EDITION, ENLARGED AT THE TJNIVEESITY PRESS [All Rights reserved,} The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001570971 PREFACE. To THE first edition of this work, published in 1867, the following was prefixed : — ' The present work was commenced in 1859, while I was a Pro- fessor of Mathematics, and far more ready at Quaternion analysis than I can now pretend to be. Had it been then completed I should have had means of testing its teaching capabilities, and of improving it, before publication, where found deficient in that respect. ' The duties of another Chair, and Sir W. Hamilton's wish that my volume should not appear till after the publication of his JSle- ments, interrupted my already extensive preparations. I had worked out nearly all the examples of Analytical Geometry in Todhunter's Collection, and I had made various physical applications of the Calculus, especially to Crystallography, to Geometrical Optics, and to the Induction of Currents, in addition to those on Kinematics, Electrodynamics, Fresnel's Wave Surface, &c., which are reprinted in the present work from the Quarterly Mathematical Journal and the Proceedings of the Royal Society of Edinburgh. ' Sir W. Hamilton, when I saw him but a few days before his death, urged me to prepare my work as soon as possible, his being almost ready for publication. He then expressed, more strongly perhaps than he had ever done before, his profound conviction of the importance of Quaternions to the progress of physical science ; and his desire that a really elementary treatise on the subject should soon be published. VI PREFACE. ' I regret that I have so imperfectly fulfilled this last request of my revered friend. When it was made I was already engaged, along with Sir W. Thomson, in the laborious work of preparing a large Treatise on Natural Philosophy. The present volume has thus been written under very disadvantageous circumstances, espe- cially as I have not found time to work up the mass of materials which I had originally collected for it, but which I had not put into a fit state for publication. I hope, however, that I have to some extent succeeded in producing a thoroughly elementary work, intelligible to any ordinary student; and that the numerous ex- amples I have given, though not specially chosen so as to display the full merits of Quaternions, will yet sufficiently shew their admir- able simplicity and naturalness to induce the reader to attack the Lectures and the Elements ; where he will find, in profusion, stores of valuable results, and of elegant yet powerful analytical investiga- tions, such as are contained in the writings of but a very few of the greatest mathematicians. For a succinct account of the steps by which Hamilton was led to the invention of Quaternions, and for other interesting information regarding that remarkable genius, I may refer to a slight sketch of his life and works in the North British Review for September 1866. ' It will be found that I have not servilely followed even so great a master, although dealing with a subject which is entirely his own. I cannot, of course, tell in every case what I have gathered from his published papers, or from his voluminous correspondence, and what I may have made out for myself. Some theorems and processes which I have given, though wholly my own, in the sense of having been made out for myself before the publication of the Elements, I have since found there. Others also may be, for 1 have not yet read that tremendous volume completely, since much of it bears on developments unconnected with Physics. But I have endeavoured throughout to point out to the reader all the more important parts of the work which I know to be wholly due to Hamilton. A great part, indeed, may be said to be obvious to any one who has mastered the preliminaries ; still I think that, in the PREFACE. VU two last Chapters especially, a good deal of original matter will be found. ' The volume is essentially a working one, and, particularly in the later Chapters, is rather a collection of examples than a detailed treatise on a mathematical method. I have constantly aimed at avoiding too great extension ; and in pursuance of this object have omitted many valuable elementary portions of the subject. One of these, the treatment of Quaternion logarithms and exponentials, I greatly regret not having given. But if I had printed all that seemed to me of use or interest to the student, I might easily have rivalled the bulk of one of Hamilton's volumes. The beginner is recommended merely to read the first five Chapters, then to work at Chapters VI, VII, VIII (to which numerous easy Examples are appended). After this he may work at the first five, with their (more difficult) Examples ; and the remainder of the book should then present no difficulty. ' Keeping always in view, as the great end of every mathematical method, the physical applications, I have endeavoured to treat the subject as much as possible from a geometrical instead of an analy- tical point of view. Of course, if we premise the properties of i,j, k merely, it is possible to construct from them the whole system* ; just as we deal with the imaginary of Algebra, or, to take a closer analogy, just as Hamilton himself dealt with Couples, Triads, and Sets. This may be interesting to the pure analyst, but it is repulsive to the physical student, who should be led to look upon i, _/, k from the very first as geometric realities, not as algebraic imaginaries. ' The most striking peculiarity of the Calculus is that muUipli- cation is not generally commutative, i.e. that qr is in general different from rq, r and q being quaternions. Still it is to be remarked that something similar is true, in the ordinary coordinate methods, of operators and functions : and therefore the student is not wholly unprepared to meet it. No one is puzzled by the fact that log.cos.a; * This has been done by Hamilton himself, as one among many methods he has employed ; and it is also the foundation of a memoir by M. AU^gret, entitled Esmi sv/r le Calcul des Quaternions (Paris, 1862). viii PREFACE. is not equal to cos.log.a?, or that v/ j- is not equal to -^s/V' Sometimes, indeed, this rule is most absurdly violated, for it is usual to taJie cos^a; as equal to (cos xY, while cos-^a; is not equal to (cos «)"■'. No such incongruities appear in Quaternions j but what is true of operators and functions in other methods, that they are not generally commutative, is in Quaternions true in the multipli- cation of (vector) coordinates. ' It will be observed by those who are acquainted with the Cal- culus that I have, in many cases, not given the shortest or simplest proof of an important proposition. This has been done with the view of including, in moderate compass, as great a variety of methods as possible. With the same object I have endeavoured to supply, by means of the Examples appended to each Chapter, hints (which will not be lost to the intelligent student) of farther develop- ments of the Calculus. Many of these are due to Hamilton, who, in spite of his great originality, was one of the most excellent examiners any University can boast of. ' It must always be remembered that Cartesian methods are mere particular cases of Quaternions, where most of the distinctive fea- tures have disappeared; and that when, in the treatment of any particular question, scalars have to be adopted, the Quaternion solution becomes identical with the Cartesian one.. Nothing there- fore is ever lost, though much is generally gained, by employing Quaternions in preference to ordinary methods. In fact, even when Quaternions degrade to scalars, they give the solution of the most general statement of the problem they are applied to, quite inde- pendent of any limitations as to choice of particular coordinate axes. 'There is one very desirable object which such a work as this may possibly fulfil. The University of Cambridge, while seeking to supply a real want (the deficiency of subjects of examination for mathematical honours, and the consequent frequent introduction of the wildest extravagance in the shape of data for " Problems "), is in danger of making too much of such elegant trifles as Trilinear PREFACE. IX Coordinates, while gigantic systems like Invariants (which, by the way, are as easily introduced into Quaternions as into Cartesian methods) are quite beyond the amount of mathematics which even the best students can master in three years' reading. One grand step to the supply of this want is, of course, the introduction into the scheme of examination of such branches of mathematical physics as the Theories of Heat and Electricity. But it appears to me that the study of a mathematical method like Quaternions, which, while of immense power and comprehensiveness, is of extraordinary sim- plicity, and yet requires constant thought in its applications, would also be of great benefit. With it there can be no " shut your eyes, and write down your equations," for mere mechanical dexterity of analysis is certain to lead at once to error on account of the novelty of the processes employed. 'The Table of Contents has been drawn up so as to give the student a short and simple summary of the chief fundamental for- mulae of the Calculus itself, and is therefore confined to an analysis of the first five [and the two last] chapters. ' In conclusion, I have only to say that I shall be much obliged to any one, student or teacher, who will point out portions of the work where a difficulty has been found ; along with any inaccuracies which may be detected.. As I have had no assistance in the revision of the proof-sheets, and have composed the work at irregular in- tervals, and while otherwise laboriously occupied, I fear it may contain many slips and even errors. Should it reach another edition there is no doubt that it will be improved in many important par- ticulars.' To this I have now to add that I have been equally surprised and delighted by so speedy a demand for a second edition — and the more especially as I have had many pleasing proofs that the work has had considerable circulation in America. There seems now at last to be a reasonable hope that Hamilton's grand in- vention will soon find its way into the working world of science, to which it is certain to render enormous services, and not be laid X PREFACE. aside to be unearthed some centuries hence by some grubbing antiquary. It can hardly be expected that one whose time is mainly en- grossed by physical science, should devote much attention to the purely analytical and geometrical applications of a subject like this ; and I am conscious that in many parts of the earlier chapters I have not fully exhibited the simplicity of Quaternions. I hope, however, that the corrections and extensions now made, especially in the later chapters, will render the work more useful for my chief object, the Physical Applications of Quaternions, than it could have been in its first crude form. I have to thank various correspondents, some anonymous, for suggestions as well as for the detection of misprints and slips of the pen. The only absolute error which has been pointed out to me is a comparatively slight one which had escaped my own notice : a very grave blunder, which I have now corrected, seems not to have been detected by any of my correspondents, so that I cannot be quite confident that others may not exist. I regret that I have not been able to spare time enough to re- write the work ; and that, in consequence of this, and of the large additions which have been made (especially to the later chapters), the whole will now present even a more miscellaneously jumbled appearance than at first. It is well to remember, however, that it is quite possible to make a book too easy reading, in the sense that the student may read it through several times without feeling those difficulties which (except perhaps in the case of some rare genius) must attend the acquisition of really useful knowledge. It is better to have a rough climb (even cutting one's own steps here and there) than to ascend the dreary monotony of a marble staircase or a well-made ladder. Royal roads to knowledge reach only the par- ticular locality aimed at — and there are no views by the way. It is not on them that pioneers are trained for the exploration of unknown regions. But I am happy to say that the "possible repulsiveness of my PREFACE. xi early chapters cannot long- be advanced as a reason for not at- tacking this fascinating subject. A still more elementary work than the present will soon appear, mainly from the pen of my colleague Professor Kelland. In it I give an investigation of the properties of the linear and vector function, based directly upon the Kineinatics of Homogeneous Strain, and therefore so different in method from that employed in this work that it may prove of interest to even the advanced student. Since the appearance of the first edition I have managed (at least partially) to effect the application of Quaternions to line, surface, and volume integrals, such as occur in Hydrokinetics, Electricity, and Potentials generally. I was first attracted to the study of Quaternions by their promise of usefulness in such applications, and, though I have not yet advanced far in this new track, I have got far enough to see that it is certain in time to be of incalculable value to physical science. I have given towards the end of the work all that is necessary to put the student on this track, which will, I hope, soon be followed to some purpose. One remark more is necessary. I have employed, as the positive direction of rotation, that of the earth about its axis, or about the sun, as seen in our northern latitudes, i.e. that opposite to the direc- tion of motion of the hands of a watch. In Sir W. Hamilton's great works the opposite is employed. The student will find no difficulty in passing from the one to the other ; but, without pre- vious warning, he is liable to be much perplexed. With regard to notation, I have retained as nearly as possible that of Hamilton, and where new notation was necessary I have tried to make it as simple and as little incongruous with Hamil- ton's as possible. This is a part of the work in which great care is absolutely necessary; for, as the subject gains development, fresh notation is inevitably required ; and our object must be to make each step such as to defer as long as possible the revolution which must ultimately come. Many abbreviations are possible, and sometimes very useful in private work ; but, as a rule, they are un suited for print. Every xii PREFACE. analyst, like every short-hand writer, has his own special con- tractions ; but, when he comes to publish his results, he ought invariably to put such devices aside. If all did not use a com- mon mode of public expression, but each were to print as he is in the habit of writing for his own use, the confusion would be utterly intolerable. Finally, I must express my great obligations to my friend M. M. U. Wilkinson of Trinity College, Cambridge, for the care with which he has read my proofs, and for many valuable sug- gestions. P. G. TAIT. College, EDnfBUROH, Octoher 1873. CONTENTS. Chapter I. — Vectoks and their Composition 1-22 Sketch of the attempts made to represent geometrically the unaginary of algebra. §§ 1-13. De Moivre's Theorem interpreted in plane rotation. § 8. Curious speculation of Servois. §11. Elementary geometrical ideas connected 'with relative position. § 15. Definition of a Vbctoe. It may be employed to denote translation. § 16. Expression of a vector by one symbol, containing implicitly three distinct numbers. Extension of the signification of the symbol = . § IS. The sign + defined in accordance with the interpretation of a rector as representing translation. § 19. Definition of - . It simply reverses a vector. § 20. Triangles and polygons of vectors, analogous to those of forces and of simul- taneous velocities. § 31. When two vectors are paralkl we have a = xp. § 22. Any vector whatever may be expressed in terms of three distinct vectors, which are not coplana, by the formula p = xa+yP+zy, which exhibits the three numbers on which the vector depends. § 23. Any vector in the same plane with a and /S may be written p = xa+yp. §24. The equation 'sr = p, between two vectors, is equivalent to three distinct equations among numbers. § 25. The Oonmmtative and Associative Laws hold in the combination of vectors by the signs + and - . § 27. The equation p = »/S, where ^ is a variable, and p a fixed, vector, represents a line drawn through the origin parallel to j3. p = O + K/S is the equation of a line drawn through the extremity of a and parallel to jS, § 28. p = ya+x§ represents the plane through the origin parallel to a and p. § 29. xiv CONTENTS. The condition that p, a, /3 may terminate in the same line is p/) + jo + r/S = 0, subject to the identical relation Similarly pp + qa + r§ + ay = 0, with p + q^ + r-\rs = 0, is the condition that the extremities of four vectors lie in one plane. § 30. Examples with solutions. § 81. Differentiation of a vector, when given as a function of one number. §§ 32-38. If the equation of a curve be p = 4>{s) where s is the length of the arc, dp is a vector tangent to the curve, and its length is ds. §§ 38, 39. Examples with solutions. §§ 40-44. Examples to Chapter 1 22-24 Chapter II. — Products and Quotients of Vectors . . . 25-46 Here we begin to see what a quaternion is. When two vectors are parallel their quotient is a number. §§ 45, 46. When they are perpendicular to one another, their quotient is a vector per- pendicular to their plane. § 47, 72. When they are neither parallel nor perpendicular the quotient in general involves fovtr distinct numbers — and is thus a Quatbbnion. § 47. A quaternion regarded as the operator which turns one vector into another. It is thus decomposable into two factors, whose order is indifferent, the stretching factor or Tenbob, and the turning factor or Versob. These are denoted by Tq, and Uq. § 48. The equation /3 = j a o gives = It or /3a~' = g, hiit not in general nr^^ = q. §49. q or j3a~' depends only on the relative lengths, and directions, of jS and a. §60. Reci^ocal of a quaternion defined, 2=-g,ves-orgi = -, y-2-' = -^. U.q-^ = {Uq)-\ §51. Definition of the Conjugate of a quaternion, Kq={Tqyqr\ and qKq = Kq.q = [Tqf. § 52. Eepresentation of versors by arcs on the unit-sphere. § 53. Versor multiplication illustrated by the composition of arcs. § 54. Proof that K{qr) = Kr . Kq. § 55. Proof of the Associative Law of Multiplication p.qr=^pq.r. §§57-60. [Digression on Spherical Conies. § 59'.] CONTENTS. XV Quaternion addition and subtraction are commutative. § 61. Quaternion multiplication and division are disti-^uiive. § 62. CompoEdtion of quadrantal veraors in planes at right angles to each other. Calling them i, j, k, we have i'=f=k^= - 1, »)■= -ji = h, jh= -hj=i, K= -ilc=j, yh=-l. §§64-71. A unit-vector, when employed as a factor, may be considered as a quadrautal versor whose plane is perpendicular to the vector, Hence the equations just written are true of any set of rectangular unit-vectors i, j, Js. § 72. The product, and the quotient, of two vectors at right angles to each other is a third perpendicular to both. Hence Ka = -a, and {Ta)^ = aKa=-a'. §73. Every versor may be expressed as a power of some unit-vector. § 74. Every quaternion may be expressed as a power of a vector. § 73. The Index Law is true of quaternion multiplication and division. § 76. Quaternion considered as the sum of a SOALAB and Yeoiob. q = ^ = x+y = Si + Vi. §77., a Proof that SKq = Sq, YKq = -7q, §79. Quadrinomial expression for a quaternion q = w+ix+jy + Jcz. An equation between quaternions is equivalent to four equations between numbers (or scalars). § 80. Second proof of the distributive law of multiplication. § 81. Algebraic determination of the constituents of the product and quotient of two vectors. §§ 82-84. Second proof of the associative law of multiplication. § 85. Proof of the formulae SajS = S^a, FajS = - rpa, o/S = K^a, S.qrs = S.rsq = S.sqr, S. a/Sv = S.pya = iS'.70jS = - S^ ayp = &c. §§ 86-89. Proof of the formulae V.aVpy = ySaP-pSya, V. 0JS7 = aSpy - pSya + 7/S0/S; 7.0/87= ^-Y/So, V. FajS Vyd = o<S'.;875 - 185.07S, = SS.aPy-yS.a^S, SS.aPy = aS.pyS + pS.yaS + yS.apS, = VapSyS+ r§ySad+ VyaSpS. §§ 90-92. Hamilton's proof that the product of two parallel vectors must be a scalar, and that of perpendicular vectors, a vector; if quaternions are to deal with space indifferently in all directions. § 93. Examples to Chaptek II 46, 47 xvi CONTENTS. Chapter III. — Interpretations and Transformations of Quaternion Expressions 48-67 If 6 be the angle between two vectors, a and j9, we have S^ = ^cos e, SaB = - TaT^ cosff, o xo a Ta Applications to plane trigonometry. §§ 94-97. shews that o is perpendicular to jS, while Fo/3 = 0, shews that a and p are parallel. S.aPy is the volume of the parallelepiped three of whose conterminous edges are a, jS, 7. Hence S.aPy = shews that a, j3, 7 are coplanar. Expression of S. apy as a determinant. §§ 98-102. Proof that {Tg)' = {Si)'+ {TVq)", and T{qr) = Tq, Tr. % 103, Simple propositions in plane trigonometry. § 104. Proof that - apa~^ ia the vector reflected ray, when j3 is the incident ray and o normal to the reflecting surface. § 105. Interpretation of 0/87 when it is a vector. § 106. Examples of variety in simple transformations. § 107. Introduction to spherical trigonometry. §§ 108-113. Bepreaentation, graphic, and by quaternions, of the spherical excess. §g 114, IIS. Loci represented by different equations — points, lines, surfaces, and solids. §§ 116-119. Proof that r-i (rV)* g-i = U(rq + KrKq). § 120. Proof of the transformation (Sv^pf + (S^pf + (Sypf = i^^^y, -''^ M«-}=^"(v55l>«-v^>7). ^121.122. BlQUATEENlONS. §§ 123-125. Convenient abbreviations of notation. §§ 126, 127. Examples to Chaptbe III 68-70 Chapter IV. — Differentiation op Quaternions .... 71-76 Definition of a dififerential, where dq is any quaternion whatever. We may write dFq =f{q, dq), where / is linear and homogeneous in dq; but we cannot generally write dFq = f{q)dq. §§128-131. CONTENTS. xvii Definition of the differential of aTTunction of more qnatemions ttan one. d(qr) = qdr + dq.r, but not generally d($r) = qdr + rdq. § 132. Proof that ^ = S^, Tp p ^=F^,&c. §133. Up p Successive differentiation; Taylor's theorem. §§ 134, 135. If the equation of a surface be -P0>) = c, the differential may be written Svdp = 0, where >< is a vector normal to the surface. § 137. Examples to Chaptbe IV 76 Chaptee V. — The Solution op Equations op the First Degree. 77-100 The most general equation of the first degree in an unknown quaternion q, may be written 2 V. aqb + S .cq = d, where a, h, c, d are given quaternions. Elimination of ;S'}, and reduction to the vector equation <1>P = S. aSPp = y. |§ 138, 139.' : -^ General proof that ^'p is expressible as a linear function of p, <pp, and <p'p. §liO. Value of <l> for an ellipsoid, employed to illustrate the general theory. §§ 141-143. Hamilton's solution of (pp = y. If we write Sa<j>p = Sp(p'a, the functions <p and ^' are said to be conjugate, and m^-^V\ii = T<t>'\<t>'p.. Proof that m, whose value may be written as S .<p'K<p! fup'v is the same for all values of \, n,v. §§ 144-146. Proof that if »n^ = m + jHi jr + 7»j jr» + j', S (\<p'iup'v + f'Xfup'v + <p'\<p'ia/) where m^ = and S.XiJiv 8 (X^0V + <l>'\iiv + K(p'nv) ' S.\iiv (which, like m, are Invarianti,) then mg (<p + g)~^ VXn = (m^-' -k-gx + f) ^^f- Also that X = '»a— ■/>> whence the final form of solution m<p-^ = m.i-mi<p + <p''. §§147,148. Examples. § 149-161. b xviii CONTENTS. The fundamental cubic (/I'-mjif' + m.^-m = (<f-£rj) (^-ffi)("?'-ff.) = 0. When is its own conjugate, the roots of the cubic are real ; and the equation ^Plip. = 0> or (.<p-g) P = 0, is satisfied by a set of three real and mutually perpendicular vectors. Geometrical interpretation of these results. §§ 162-166. Proof of the transformation i>p =fp + hV. (i + ek)'f (i—ek) where (<^— ffi)* = 0, C — ) Another transformation is (pp = aaVap + bPSPp. §§167-169. Other properties of i^. Proof that Sp(<t> + g)~*P = 0) and Sp (<p + h)~^p = represent the same surface if mSp(p~^p = ghp^. Proof that when ip is not self-conjugate ipp = (p'p + Vep. Proof that, if q = a(pa + 0(p0 + 71^7, where a, P, 7 are any rectangular unit-vectors whatever, we have Sq— — TOj, Vq = f. This quaternion can be expressed in the important form 2 = v#. §§ 170-174. Degrees of indeterminateness of the solution of a quaternion equation — Examples. §§ 176-179. The linear function of a quaternion is given by a symbolical biquadratic. §180. Particular forms of linear equations. §§ 181-183. A quaternion equation of the mth degree in general involves a scalar equation of degree m*. § 184. Solution of the equation ^ = qa + T>. §185. Examples to ChaptebV 101-103 Chapter VI. — Gbometei of the Straight Line and Plane , 104-117 Examples to Chafteb VI 117-119 Chapter VII. — ^The Sphere and Cyclic Cone . . . 120-132 EXAMPLES TO Chapteb VII 132-134 Chapter VIII. — Surfaces of the Second Order . . 135-151 Examples xo Chapter VIII 151-154 CONTENTS. XIX Chapter IX. — Geometry op (?urves and Surfaces . 155-186 Examples to Chapter IX 187-194 Chapter X. — Kinematics 195-218 If p = 0< be the vector of a moving point in terms of the time, p is the vector velocity, and p the vector acceleration. (T = p = ((>'(<) is the equation of the Hodograph. p = vp' + v'p" gives the normal and tangential accelerations. Vpp = if acceleration directed to a point, whence Tpp = y. Examples. — Planetary acceleration. Here the equation is /.Dp given Vp^ — y ; whence the hodograph is p = ty~'^—iiUp.y~\ and the orbit is the section of j«r/. = Sf(7='£->-p) by the plane Syp = 0. Epitrochoids, &c. §| 336-348. Rotation of a rigid system. Composition of rotations. The operator 5s( )q—^ turns the system it is applied to through 2 b times the angle of g, about the axis of q. If the position of a system at time t is derived from the initial position by j ( ) 2~', the instantaneous axis is € = 2Vqg-^. §§ 349-359. Homogeneous strain. Criterion of pure strain. Separation of the rotational jrom the pure part. Extraction of the square root of a strain. A strain ^ is equivalent to a, pure strain V*^'^ followed by the rotation — - — . Simple Shear. §§ 360-367. '^'P''!' Displacements of systems of points. Consequent condensation and rotation. Preliminary about the use of V. §§ 368-371. Moment of inertia. § 372. Examples to Chapter X 218-221 Chapter XI. — Physical Applications 222-288 Condition of equilibrium of -a rigid system is 'SS.PSa = 0, where j8 is a vector force, a its point of application. Hence the usual six equations in the form 2j8 = 0, SVa0 = 0. Central axis, &c. §§ 373-378. For the motion of a rigid system SS(md-P)Sa = 0, whence the usual forms. The equation 2j= q<p-^{q-^yq), where y is given in terms of t and q if forces act, but is otherwise constant, contains the whole theory of the motion of a rigid body with one point fixed. Reduction to the ordinary form dt dm dx _dy _ dz "2 W~X~T~ ~Z' Here, if no forces act, W, X, T, Z are homogeneous functions of the third degree in w, x, y, z. Equation for precession. §§ 379-401. General equation of motion of simple pendulum. Foucault's pendulum. §§ 402-406. b3 CONTENTS. Problem on reflecting surfaces. § 406. Freanel's Theory of Double Eefraction. Various fonns of the equation of Fresnel's Wave-surface ; S.p(.f-p»)-V = -l. T(p-'-<l>-')-ip = 0, l:=-pp' + (,T±S)VKpViip, The conical cusps and circles of contact. Lines of vibration, &c. §§ 407-427. Electrodynamics. The vector action of a closed circuit on an element of current o, is proportional to Vai0 where ^ rVada CdUa the integration extending round the circuit. Mutual action of two closed circuits, and of solenoids. Mutual action of magnets. Potential of a closed circuit. Magnetic curves. §§ 428-448. Physical applications of , d . d , i dx dy dz Effect of V on various functions of p. = kc +jy + kz. 2 V/)=-3, VTp=V'p, vTJp = —~. V8ap=-a, v7ap=2a. Applications of the theorem S.SpV^ = SS.oV^ . §§ 449-457. Jp» Ip Farther examination of the use of V as applied to displacements of groups of points. Proof of the fundamental theorem for comparing an integral over a closed surface with one through its content ///S.V<rd^=//S.aUvd8. Hence Green's Theorem. Limitations and ambiguities. §§ 458-476. Similar theorem for double and single integrals fS.adp =//S.UV7ads. Applications of these to distributions of magnetism, and to Ampere's Directrice. Also to the Stress-function. §5 477-491. e-S<rV/(p)= f(p + <r). Applications and consequences. Separation of symbols of operation, and their treatment as quantities. §§ 492-495. Applications of V in connection with the Calculus of Variations. If A =/QTdp, SA=0 gives ^(,Qp')-vQ = 0. Ui8 Applications to Varying Action, Brachistochrones, Catenaries. §§ 496-504. Thomson's Theorem that there is one and but one solution of S.VCe'Vit) = 4irr. §505. MiSCELLANEOtrS EXAMPLES 288-296 ERRATUM. Page 102, line 20, for ^p—tpipp read <j>if/p—</«l>'p. QUATERNIONS. CHAPTER I. VECTORS, AND THEIR COMPOSITION. 1,] For more than a century and a half the geometrical re- presentation of the negative and imaginary algebraic quantities, — 1 and a/— 1, or, as some prefer to write them, — and — *, has been a favourite subject of speculation with mathematicians. The essence of almost all of the proposed processes consists in em- ploying such expressions to indicate the direction, not the length, of lines. 2.] Thus it was long ago seen that if positive quantities were measured o£F in one direction along a fixed line, a useful and lawful convention enabled us to express negative quantities of the same kind by simply laying them off on the same line in the opposite direction. This convention is an essential part of the Cartesian method, and is constantly employed in Analytical Geometry and Applied Mathematics. 3.] WaUis, towards the end of the seventeenth century^ proposed to represent the impossible roots of a quadratic equation by going out of the line on which, if real, they would have been laid off. His construction is equivalent to the consideration of v — 1 as a directed unit-line perpendicular to that on which real quantities are measured. 4. J In the usual notation of Analytical ^Geometry of two dimensions, when rectangular axes are employed, this amounts to reckoning each unit of length along Oy as +v— 1, and on Oy' as — V — 1 ; whUe on Ox each unit is +1, and on Oaf it is B 2 QUATEKNIONS. [5. — 1 . If we look at these four lines in circular order, i. e. in the order of positive rotation (opposite to that of the hands of a watch), they give r—r _ _ y3"i In this series each expression is derived from that which precedes it by multiplication by the factor v— 1. Hence we may consider -v/— 1 as an operator, analogous to a handle perpendicular to the plane of ay, whose effect on any line is to make it rotate (positively) about the origin through an angle of 90°. 5.] In such a system^ a point is defined by a single imaginary expression. Thus a + b v — 1 may be considered as a single quan- tity, denoting the point whose coordinates are a and b. Or, it may be used as an expression for the line joining that point with the origin. In the latter sense, the expression a + b \/—l implicitly contains the direction, as well as the length, of this line ; since, as we see at once, the direction is inclined at- an angle tan^^- to the axis oi X, and the length is \/a^ + J^. 6.] Operating on this symbol by the factor V— 1, it becomes — 3-|-a\/— 1 ; and now, of course, denotes the point whose x avAy coordinates are —b and a ; or the line joining this point with the origin. The length is still Va^ + b"^, but the angle the line makes with the axis of a; is tan~^ (~ 7") ' 'w^^'ich is evidently 90° greater than before the operation. 7.3 De Moivre's Theorem tends to lead us still farther in the same direction. In fact, it is easy to see that if we use, instead of >/— 1, the more general factor cosa+ ^/— 1 sin a, its effect on any line is to turn it through the (positive) angle a in the plane oix,y. [Of course the former factor, 'J —I, is merely the par- ticular case of this, when a = - •! 2 -■ Thus (cos a -I- \/ — 1 sina) (a + ^-s/— 1) = a cos o — 5 sina-l- \/— 1 (asino + J cos a), by direct multiplication. The reader will at once see that the new form indicates that a rotation through an angle a has taken place, if he compares it with the common formulae for turning the co- ordinate axes through a given angle. Or, in a less simple manner, thus — Length =\/(a coso— 6sina)^ + (asina + 5cosa)^ = \/a'^ -I- b'^ as before. 12.] VECTORS, AND THEIR COMPOSITIO^^ 3 Inclination to axis of a; j , tan a-\ — , _, « sin a + cos a , , a = tan '■ j-^ — = tan-i = a cos a— sin a o I 1 tana = a + tan~i - • a 8.] We see now, as it were, wA^ it happens that (cos a 4- V — 1 sin a)™ = cos »ia + /^/^^^ sin ma. In fact, the first operator produces m successive rotations in the same direction, each through the angle a ; the second, a single rotation through the angle ma. 9.] It may be interesting, at this stage, to anticipate so far as to state that a Quaternion can, in general, he put under the form N {cos d + -ay sin 6), where iV" is a numerical quantity, 8 a real angle, and This expression for a quaternion bears a very close analogy to the forms employed in De Moivre's Theoreili ; but there is the essential difference (to which Hamilton's chief invention referred) that -sr, is not the algebraic v — 1, but may be an^ directed unit-line what- ever in space. 10.] In the present century Argand, Warren, and others, extended the results of WalHs and De Moivre. They attempted to express as a line the product of two lines each represented by a symbol such as a + J v^— 1. To a certain extent they succeeded, but sim- plicity was not gained by their methods, as the terrible array of radicals in Warren's Treatise suflBciently proves. 11.] A very curious speculation, due to Servois and published in 1813 in Gergonne's Annates, is the only one, so far as has been discovered, in which the slightest trace of an anticipation of Quaternions is contained. Endeavouring to extend to space the form a + J\/— 1 for the plane, he is guided by analogy to write for a directed unit-line in space the form p cos a + §■ cos /3 + r cos y, where a, p, y are its inclinations to the three axes. He perceives easily that j9, q, r must be non-reals : but, he asks, " seraient-elles imaginaires reductibles a la forme generale A-\-B '^ — I ?" This he could not answer. In fact they are the i, j, k of the Quaternion Calculus. (See Chap. II.) 12.] Beyond this, few attempts were made, or at least recorded, in earlier times, to extend the principle to space of three dimensions ; B a 4 QUATERNIONS. [ 1 3- and, though many such have been made within the last forty years, none, with the single exception of Hamilton's, have resulted in simple, practical methods; all, however ingenious, seeming to lead at once to processes and results of fearful com- plexity. For a lucid, complete, and most impartial statement of the claims of his predecessors in this field we refer to the Preface to Hamilton's Lectures on Quaternions. 13.] It was reserved for Hamilton to discover the use of -n/ — 1 as a geometric realitij, tied down to no particular direction in space, and this use was the foundation of the singularly elegant, yet enormously powerful, Calculus of Quaternions. While all other schemes for using -s/^ to indicate direction make one direction in space expressible by real numbers, the re- mainder being imaginaries of some kind, leading in general to equations which are heterogeneous ; Hamilton makes all directions in space equally imaginary, or rather equally real, thereby ensuring to his Calculus the power' of dealing with space indifferently in all directions. In fact, as we shall see, the Quaternion method is independent of axes or any supposed directions in space, and takes its reference lines solely from the problem it is applied to. 14.] But, for the purpose of elementary exposition, it is best to begin by assimilating it as closely as we can to the ordinary Cartesian methods of Geometry of Three Dimensions, which are in fact a mere particular case of Quaternions in which most of the distinctive features are lost. We shall find in a little that it is capable of soaring above these entirely, after having employed them in its establishment; and, indeed, as the inventor's works amply prove, it can be established, ah initio^ in various ways, without even an allusion to Cartesian Geometry. As this work is written for students acquainted with at least the elements of the Cartesian method, we keep to the first-mentioned course of exposition ; especially as we thereby avoid some reasoning which, though rigorous and beautiful, might be apt, from its subtlety, to prove repulsive to the beginner. We commence, therefore, with some very elementary geometrical ideas. 15.] Suppose we have two points A and B in space, and suppose A given, on how many numbers does ^'s relative position depend ? If we refer to Cartesian coordinates (rectangular or not) we find 1 9-] VECTOES, AND THEIR COMPOSITION. 5 • that the data required are the excesses of ^'s three coordinates over those of A. Hence three numbers are required. Or we may take polar coordinates. To define the moon's position with respect to the earth we must have its Geocentric Latitude and Longitude, or its Right Ascension and Declination, and, in addition, its distance or radius-vector. Three again. 16.J Here it is to be carefully noticed that nothing has been said of the actual coordinates of either A or B, or of the earth and moon, in space ; it is only the relative coordinates that are contemplated. Hence any expression, as AB, denoting a line considered with reference to direction as well as length, contains implicitly three numbers, and all lines parallel and equal to AB depend in the same way upon the same three. Hence, all lines which are equal and parallel may he represented hy a common symbol, and that symbol contains three distinct numbers. In this sense a line is called a VEOTOE, since by it we pass from the one extremity, A, to the other, B ; and it may thus be considered as an instrument which carries A to B : so that a vector may be employed to indicate a definite translation in space. 17.] We may here remark, once for all, that in establishing a new Calculus, we are at liberty to give any definitions whatever of our symbols, provided that no two. of these interfere with, or contradict, each other, and in doing so in Quaternions simplicity and (so to speak) naturalness were the inventor's aim. 18.] Let AB be represented by a, we know that a depends on three separate numbers. Now if CD be equal in length to AB and if these lines be parallel, we have evidently CI) = AB = a, where it will be seen that the sign of equality, between vectors contains implicitly equality in length and parallelism in direction. So far we have extended the meaning of an algebraical symbol. And it is to be noticed that an equation between vectors, as a = /3, contains three distinct equations between mere numbers. 19.] We must now define + (and the meaning of — will follow) in the new Calculus. Let A, B, C be any three points, and (with the above meaning of =) let AB=a, BG=I3, AC=y. If we define + (in accordance with the idea (§ 16) that a vector represents a translation) by the equation 6 QUATERNIONS. [20. or AB + £C = AC, we contradict nothing that precedes, but we at once introduce the idea that vectors areata he compounded, in direction and magnitude, like simultaneous velocities. A reason for this may be seen in another way if we remember that by adding the diflferences of the Cartesian coordinates of A and B, to those of the coordinates of B and C, we get those of the coordinates of A and C. Hence these coordinates enter" linearly into the expression for a vector. 20.] But we also see that if C and A coincide (and C may be any point) AQ = 0, for no vector is then required to carry A to C. Hence the above relation may be written, in this case, AB+BA = 0, or, introducing, and by the same act defining, the symbol — , BA=-AB. Hence, t/ie symbol — , applied to a vector, simply shows that its direction is to he reversed. And this is consistent with all that precedes ; for instance, ab+bc = m;, and AB = AC-BC, or =AG+CB, are evidently but different expressions of the same truth. 21.] In any triangle, ABC, we have, of course, IB + BC+CA^ 0; and, in any closed polygon, whether plane or gauche, AB-k^BC+ '. + TZ+ZA = 0. In the case of the polygon we have also AB + BC+ + fZ=AZ. These are the well-known propositions regarding composition of velocities, which, by the second law of motion, give us the geo- metrical laws of composition of forces. 22.] If we compound any number of parallel vectors, the result is obviously a numerical multiple of any one of them. Thus, if A, B, C are in one straight line, BC=i>!AB; where a; is a number, positive when B lies between A and C, other- wise negative : but such that its numerical value, independent of sign, is the ratio of the length of BC to that of AB. This is 25-] VECTOES, AND THEIR COMPOSITION. 7 at oace evident if AB and BC be commensurable j and is easily extended to incommensurables by the usual reductio ad absurdum. 23.] An important, but almost obvious, proposition is that any vector may he resolved, and in one way only, into three components parallel respectively to any three given vectors, no two of which are parallel, and which are not parallel to one plane. Let OA, OB, OC be the three fixed vectors, c OP any other vector. From P draw PQ parallel to CO, meeting the plane BOA in Q. [There must be a definite point Q, else PQ, and therefore CO, would be parallel to BOA, a case specially excepted.] Prom Q draw QB parallel to BO, meeting OA in B. Then we have OP = 0^ + ^ + QP (§ 21), and these components are respectively parallel to the three given vectors. By § 22 we may express OB as a numerical multiple of OA, RQ oi OB, and QP of OC. Hence we have, generally, for any vector in terms of three fixed non-coplanar vectors, a, /3, y, OP = p = xa + yl3 + zy, which exhibits, in one form, the three numbers on which a vector depends (§ 16). Here x, y, z are perfectly definite, and can have but single values. 24.] Similarly any vector, as OQ, in the same plane with OA and OB, can be resolved into components OB, RQ, parallel re- spectively to OA and OB ; so long, at least, as these two vectors are not parallel to each other. 25.] There is particular advantage, in certain cases, in employ- ing a series of three mutually perpendicular unit-vectors as lines of reference. This system Hamilton denotes by i,j, h. Any other vector is then expressible as p = xi-\-yj-\-zh. Since i, j, k are unit- vectors, x, y, z are here the lengths of con- terminous edges of a rectangular parallelepiped of which p is the vector-diagonal ; so that the length of p is, in this case, Let TO- = ^i + T/y+C^ be any other vector, then (by the proposition of § 23) the vector equation p =. 'ss obviously involves the following three equations among numbers, « = ^. y = ■<), z=C 8 QUATERNIONS. [26. Suppose i to be drawn eastwards, J northwards, and k upwards, this is equivalent merely to saying that if two points coincide, ihey are equally to tie east {or west) of any third point, equally to the north {or south) of it, and equally elevated above {or depressed below) its level. 26.] It is to be carefully noticed that it is only when a, fi, y are not coplanar that a vector equation such as p = OT, or «a-f ^;3 + «y = £o + jj/3+Cy, necessitates the three numerical equations m = i, y = n, « = ^ For, if a, ^j y be coplanar (§ 24), a condition of the following form must hold y = aa + b^. Hence p = {x + za)a+{y + zh)p, ^={i+Ca)a + {r, + Cb)^, and the equation p ■= m now requires only the two numerical conditions x + za= ^+Ca, y + zb = r] + (b. 27.] The Commutative and Associative Laws hold in the combination of vectors by the signs + and — . It is obvious that, if we prove this for the sign + , it will be equally proved for — , because — before a vector (§ 20) merely indicates that it is to be reversed before being considered positive. Let A, B, G, B be, in order, the corners of a parallelogram ; we have, obviously, Jb = SC, IT) = BG. And AB + BC= IC=An+BC=BC+AB. Hence the commutative law is true for the addition of any two vectors, and is therefore generally true. Again, whatever four points are represented by A, B, C, J), we have lD = IB+BB = AC-\-UD, or substituting their values for AB, BB, AC respectively, in these three expressions, lB+BC-\-CB^AB + {BC+CB)= {AB + BC) + CB. And thus the truth of the associative law is evident. 28.] The equation „ — xB, where p is the vector connecting a variable point with the origin, /3 a definite vector, and x an indefinite number, represents the straight line drawn from the origin parallel to ^ (§ 22). 30.J VECTOES, AND THEIR COMPOSITION. 9 The straight line drawn from A, where OA = a, and parallel to j8, has the equation p = a + a;/3 ; (1) In words, we may pass directly from to P by the vector OP or p ; or we may pass first to A, by means of OA or a, and then to P along a vector parallel to ^ (§ 16). Equation (1) is one of the many useful forms into which Quater- nions enable us to throw the general equation of a straight line in space. As we have seen (§ 25) it is equivalent to three numerical equations ; butj as these involve the indefinite quantity x, they are virtually equivalent to but two, as in ordinary Geometry of Three Dimensions. 29.] A good illustration of this remark is furnished by the fact that the equation p = va + se^, which contains two indefinite quantities^ is virtually equivalent to only one numerical equation. And it is easy to see that it re- presents the plane in which the lines o and fi lie ; or the surface which is formed by drawing, through every point of OA, a line parallel to OB. In fact, the equation, as written, is simply § 24 in symbols. And it is evident that the equation P = y+ya + oo^ is the equation of a plane passing through the extremity of y, and parallel to a and ;8. It will now be obvious to the reader that the equation P =i'i«i+i'2a2+ = '2pa, where a^, Og, &c. are given vectors, and Pi,P2> ^c. numerical quan- tities, represents a strd/igM line if i5i,j»2J &c. be linear functions of one indeterminate number ; and a plane, if they be linear expres- sions containing two indeterminate numbers. Later (§31 {})), this theorem will be much extended. Again, the equation p = xa + y^-^zy refers to any point whatever in space, provided a, /3, y are not coplanar. {Ante, § 23). 30.] The equation of the line joining any two points A and B, where OA = a and OB = 13, is obviously P = a + a;(/3— a), or p = ^ + y(a-/3). These equations are of course identical, as may be seen by putting 1—y for ss. 10 QUATERNIONS. [3 1. The first may be written p + (x—l)a — x^ = ; or j)p + qa + rfi = 0, subject to the condition p + q + r = identically. That is — A homogeneous linear function of three vectors, equated to zero, expresses that the extremities of these vectors are in one straight line, if the sum of the coefficients he identically zero. Similarly, the equation of the plane containing the extremities A, B, C of the three non-coplanar vectors a, j3, y is p = a + a:(/3-a) + y(y-/3), where x and y are each indeterminate. This may be written pp + qa + r^ + sy = 0, with the identical relation p + q + r + s = 0. which is the condition that four points may lie in one plane. 31.] We have already the means of proving, in a very simple manner, numerous classes of propositions in plane and solid geo- metry. A very few examples, however, must suflSce at this stage ; since we have hardly, as yet, crossed the threshold of the subject, and are dealing with mere linear equations connecting two or more vectors, and even with them we are restricted as yet to operations of mere addition. We will give these examples with a painful minute- ness of detail, which the reader will soon find to be necessary only for a short time, if at all. (a.) The diagonals of a parallelogram bisect each other. Let ABGB be the parallelogram, the point of intersection of its diagonals. Then iO + ^= IB =^G = Bb+OC, which gives AO-OC = BO-OB. The two vectors here equated are parallel to the diagonals respect- ively. Such an equation is, of course, absurd unless (1) The diagonals are parallel, in which case the figure is not a parallelogram ; (2) Jo = Oa, and ^ = OB, the proposition. (i.) To show that a triangle can he constructed, whose sides are parallel, and equal, to the hisectors of the sides of any triangle. Let ABC be any triangle, Aa, Bh, Cc the bisectors of the sides. 3I-] VECTORS, AND THEIR COMPOSITION. 11 _• Then Aa, = AB+Ba = AB^-\ BC, Bb - - - =BC+hCA, Co - - - =Cl+\lB. Hence Aa + Bb + Cc=^{lB + BG+CA)^(); which (§21) proves the proposition. Also Aa — JS+\BC = AB-\{Cl+AB) = ^{AB-ai) = i{lB+IC), results which are sometimes useful. They may be easily verified by producing Aa to twice its length and joining the extremity with B. {¥.) The bisectors of the sides of a triangle meet in a point, which trisects each of them. Taking A as origin, and putting o, /3, y for vectors parallel, and equal, to the sides taken in order BC, CA, AB; the equation of Bb is (§ 28 (1)) p = y + «(y+f) = (i+a')y + |^- That of Cc is, in the same way, p = -(l + y)^-|y. At the point 0, where Bb and Cc intersect, p=(H-a;)y+-/3=-(l+j.)^-|y. Since y and ^ are not parallel, this equation gives H-«' = -f, and | = _(i+y). From these a; = y = — |. Hence iO = 4 (y-/3) = 1 2a. (See Ex. («).) This equation shows, being a vector one, that Aa passes through 0, and that JO : Oa : : 2: 1. (c) If 02 = a, ^ OG=aa+b^, be three given co-planar «^ ^ ^^"^^^ / \ ^^^/s^~-~^ vectors, and the lines in- ^^ fli dicated in the figure be drawn, the points «i,*i,Ci lie in a straight line. 12 QUATERNIONS. [31. We see at once, by the process indicated in § 30, that Oc = aa + b^ Qj ^ "-"■ a + b Hence we easily find -a — 2b ^ 0-a= '^ \—a Oa^ =■ Oc, = — 7 ^ b—a \—a-2b •■ l — 2a-b' These give -{l-a-2b)Oai+{l-2a-6)Obi-{b-a)Oc^ = 0. But ■ _(l-a-2i) + (l-2a-5)-(5-a) = identically. This, by § 30, proves the proposition. (d.) Let 02 = a, OB = /3, be any two vectors. If MP be parallel to OB; and OQ, BQ, be drawn parallel to AP, OP respectively; the locus of Q is a straight line parallel to OA. Let OM = ea. Then_ AP = e— la + a;)3. Hence the equation of OQis p = y(e-la+»i3); and that of ^Q is p = ^ + z{ea+co^). At Q we have, therefore, xy = l+zx, \ y{e-\) = ze.\ These give xy = e, and the equation of the locus of Q is P = «/3 + /a, i. e. a straight line parallel to OA, drawn through N in OB pro- duced, so that ON -.OB:: OM: OA. CoE. li BQ meet MP in q,P'c[=^; and if AP meet NQ in p, Qp=a. _ _ Also, for the point B we have pB = AP, Q,R=Bq. Hence, if from any two points, A and B, lines be drawn intercepting a given length Pq on a given line Mq; and if, from B their point of intersection, Bp be laid off = PA, and BQ = qB ; Q and p lie on a fixed straight line, and the length of Qp is constant. (e.) To find the centre of inertia of any system. If OA = a, OB = a^, be the vector sides of any triangle, the vector from the vertex dividing the base AB in C so that 31.] VECTORS, AND THEIE COMPOSITION. 13 BG : CA ■.:m:m-^ is For A£ is Oj — a, and therefore AC is -(oj— a). Hence 00: OA + AC = a + («! — a) .,i + Ml ma + mj^a^ This expression shows how to find the centre of inertia of two masses; m at the extremity of a, m^ at that of a^. Introduce Wg at the extremity of 02, then the vector of the centre of inertia of the three is, by a second application of the formula, »i4-««j M a + »Zi Oj^ + »«2 02 ■m-\-m■^^-\-m^ (m+mj^ + m^ For any number of masses, expressed generally by m at the extre- mity of the vector a, we have the vector of the centre of inertia '^ s(ot) ■ This may be written 2m(a—fi) = 0. Now Oj— /3 is the vector of % with respect to the centre of inertia. Hence the theorem, ^ the vector of each element of a mass, drawn from the centre of inertia, he increased in length in proportion to the mass of the element, the sum of all these vectors is zero. (_/.) We see at once that the equation where t is an indeterminate number, and a, j8 given vec- tors, represents a parabola. The origin, 0, is a point on the curve, /3 is parallel to the axis, i. e. is the diameter OB drawn from the origin, and a is OA the tangent at the origin. qp = at, 6q = 14 QUATERNIONS. [31. The secant joining the points where t has the values t and If is represented by the equation , = a. + ^+.(ar+^-a.-^) (§30) Put 1f=.t, and write x for x{if—t) [which may have any value] and the equation of the tangent at the point {t) is Put X = —t, p = — > or the intercept of the tangent on the diameter is —the abscissa of the point of contact. Otherwise: the tangent is parallel to the vector a+fit or at + pt'' OT at + if. + if. ov 0Q+ UP. BuifF=fd + 6P, hence TO = OQ. •{ff.) Since the equation of any tangent to the parabola is p = at + ^ + x{a+lii), let us find the tangents which can be drawn from a given point. Let the vector of the point be p=pa + ql3 (§24). Since the tangent is to pass through this point, we have, as con- ditions to determine t and x, i + x = p, -j + xt = q; by equating respectively the coefficients of a and /3. Hence ^ =jo+ \/^^ — 2$'. ThuSj in general, two tangents can be drawn from a given point. These coincide if ^2 _ 2^ ; that is, if the vector of the point from which they are to be drawn is „ P^ „ p =pa + qfi =Pa.+ ^p, i. e. if the point lies on the parabola. They are imaginary if 2q>p^, i. e. if the point be r being positive. Such a point is evidently within the curve, as at 72, where OQ =^/3, QP=pa, PB = r^. 3I-] VECTORS, AND THEIB COMPOSITION. 15 (A.) Calling the values o{ t f9r the two tangents found in (^) ti and ^2 respectively, it is obvious that the vector join- ing the points of contact is "2 2 which is parallel to f ^^ a + ^-i^; or, by the values of t^ and t^ in (ff), a+jB/3. Its direction, therefore, does not depend on q. In words, If pairs of tangents he ckamn to a parabola from points of a diameter pro&uced, the chords of contact are parallel to the tangent at the vertex of the diameter. This is also proved by a former result, for we must have OT for each tangent equal to Q 0. {i.) The equation of the chord of contact, for the point whose vector is p=pa + ql3, Bt ^ is thus P = a^i+^ + ^(«+J0;8). Suppose this to pass always through the point whose vector is p = aa + b^. Then we must have , , h+^ = «. ) t ^ [ or ti=p±^p^ — 2pa + 2b. Comparing this with the expression in {g), we have q =pa—h; that is, the point from which the tangents are drawn has the vector p =pa + {pa—b)^ = —b^+p{a + aj3), a straight line (§ 28 (1)). The mere form of this expression contains the proof of the usual properties of the pole and polar in the parabola ; but, for the sake of the beginner, we adopt a simpler, though equally general, process. Suppose a = 0. This merely restricts the pole to the particular diameter to which we have referred the parabola. Then the pole is Q, where p = b^; and the polar is the line TU, for which p = -6fi+pa. 16 QTTATEKNIONS. [3I. Hence the polar of any point is parallel to the tangent at the extremity of the diameter on which the point lies, and its intersection with that diameter is as far beyond the vertex as the pole is within, and vice versa. (J.) As another example let us prove the following theorem. Jf a triangle he inscribed in a parabola, tlie three points in which the sides are met by tangents at the angles lie in a straight line. Since is any point of the curve, we may take it as one corner of the triangle. Let t and ^j determine the others. Then, if OTj, OTj, iirg represent the vectors of the points of intersection of the tangents with the sides, we easily find tt. ''' = t:vt"" These values give itj^-t 2t—t-i^ t^-t^ Ai "^h-t ^t-h ty-^ „ •. ^. 1, Also — i — -i- — i- — = identically. Hencoj by § 30, the proposition is proved. ijc) Other interesting examples of this method of treating curves will, of course, suggest themselves to the student. Thus p = a cos if 4- ^ sin ^ or p = oa;+^^/l— jc^ represents an ellipse, of which the given vectors a and /3 are semi- conjugate diameters. Agam, p = aif + - or p = a tana;+^cota; evidently represents a hyperbola referred to its asymptotes. But, so far as we have yet gone with the explanation of the calculus, as we are not prepared to determine the lengths or in- clinations of vectors, we can investigate only a very small class of the properties of curves, represented by such equations as those above written. 33-] VECTORS, AND THEIR COMPOSITION. 17 (I.) We may now, in extensi^ of the statement in § 29, make the obvious remark that p = Sj)a is the equation of a curve in space, if the numbers i'i,i»2> ^^- ^^^ functions of one indeterminate. In such a case the equation is sometimes written . _ j^/f. But, if jOi, j»2J ^c. be functions of two indeterminates, the locus of the extremity of p is a surface ; whose equation is sometimes written p = <t>{t,u). (m.) Thus the equation p = acost + ^sint+yt belongs to a helix. Again, p=pa + q^ + ry with a condition of the form ap^ 4 6q^ + cr^ ■= I belongs to a central surface of the second order, of which a, p, y are the directions of conjugate diajneters. If a, b, c be all positive, the surface is an ellipsoid. 32.] In Example (_/) above we performed an operation equi- valent to the differentiation of a vector with reference to a single numerical variable of which it was given as an explicit function. "As this process is of very great use, especially in quaternion investi- gations connected with the motion of a particle or point ; and as it will afford us an opportunity of making a preliminary step towards overcoming the novel difficulties which arise in quaternion differen- tiation; we will devote a few sections to a more careful exposition of it. 33.] It is a striking circumstance, when we consider the way in which Newton's original methods in the Differential Calculus have been decried, to find that Hamilton was obliged to employ them, and not the more modern forms, in order to overcome the characteristic difficulties of quaternion differentiation. Such a thing as a differential coefficient has absolutely no meaning in quaternions, except in those special cases in which we are dealing with degraded quaternions, such as nvmibers, Cartesian coordinates, &c. But a quaternion expression has always a differential, which is, simply, what Newton called sifluadon. As with the Laws of Motion, the basis of Dynamics, so with the foundations of the Differential Calculus ; we are gradually coming to the conclusion that Newton's system is the best after all. c 18 QtTATERNIONS. [34- 34.J Suppose p to be the vector of a curve in space. Then, generally, p may be expressed as the sum of a number of terms, each of which is a multiple of a given vector by a function of some one indeterminate; or, as in § 31 (1), if P be a point on the curve, 6P=p = 4>{t). And, similarly, if Q be ani/ other point on the curve, where htis any number whatever. The vector-chord PQis therefore, rigorously, 6p = pi-p = (f>{t + bt)-cl>t. 35.] It is obvious that, in the present case, because the vectors involved in (j) are constant, and their numerical multipliers alone vary, the expression i^it^ht) is, by Taylor's Theorem equivalent to '^^*^^-ir^*^~d(^~^^ Hence, ^<^(0 ,,^^'<^ W (8^)' ^^ And we are thus entitled to write, when ht has been made inde- finitely small, , , ,,. ^ ' ,lp. dp dc\,{t) _ , In such a case as this, then, we are permitted to differentiate, or to form the differential coefficient of, a vector, according to the ordinary rules of the Differential Calculus. But great additional insight into the process is gained by applying Newton's method. 36.] Let OP be _ P = <i>[t), and OQi p= 4>{t + dt), where dt is any number whatever. The number t may here be taken as representing time, i. e. we may suppose a point to move along the curve in such a way that the value of t for the vector of point P of the curve denotes the interval which has elapsed (since a fixed epoch) when the moving point has reached the extremity of that vector. If, then, dt represent any interval, finite or not, we see that 6q^=^{t+dt) will be the vector of the point after the additional interval dt. 38.J YECTORS, AND THEIR COMPOSITION. 19 But this, in general, gives us little or no information as to the velocity of the point at P. We shall get a better approximation by halving the interval di, and finding Q^, where 0^2= <^ {i+h ^i)) as the position of the moving point at that time. Here the vector virtually described in ^df is PQ^. To find, on this supposition, the vector described in di, we must double PQ2) and we find, as a second approximation to the vector which the moving point would have described in time dt, if it had moved for that period in the direction and with the velocity it had at P, Tq2=2PQ^ = 2{0Q2-6P) = 2{(l>{i+kdt)-(l){t)}. The next approximation gives P^, = 3PQ,= 3{6Q,-6P) = 3{4>{i+idt)-<t>{i)]. And so on, each step evidently leading us nearer the sought truth. Hence, to find the vector which would have been described in time dt had the circumstances of the motion at P remained undisturbed, we must find the value of dp = Tq = J^:,=^ai^cj>(t + ^dt)-<j>{t)\- We have seen that in this particular case we may use Taylor's Theorem. We have, therefore, dp = J^,=^ X \^'{t)\ di+<j>"{t) ^ ^ 4 &c. I = 4)' (t) dt. And, if we choose, we may now write 37.] But it is to be most particularly remarked that in the whole of this investigation no regard whatever has been paid to the magnitude of dt. The question which we have now answered may be put in the form — A point describes a given curve in a given manner. At any point of its path its motion suddenly ceases to he accelerated. What space will it describe in a definite interval ? As Hamilton well observes, this is, for a planet or comet, the case of a ' celestial Atwood's machine.' 38.] If we suppose the variable, in terms of which p is expressed, to be the arc, s, of the curve measured from some fixed point, we find as before , ,,,,, ,, ^'{t)ds dp = ^{t)dt = ^^-^ = 4>'{s}ds. ^^ C 3 20 QUATERNIONS. [39. From the very nature of the question it is obvious that the length of dp must in this case 'be ds. This remark is of importance, as we shall see later ; and it may therefore be useful to obtain afresh the above result without any reference to tiine or velocity. 39.] Following strictly the process of Newton's Vllth Lemma, let us describe on Pq^ an arc similar to PQg, and so on. Then obviously, as the subdivision of ds is carried farther, the new arc (whose length is always ds) more and more nearly coincides with the line which expresses the corresponding approximation to <?p. 40.] As a final example let us take the hyperbola Here dp = {a-^)dt. This shews that the tangent is parallel to the vector at --• In words, if the vector {from the centre) of a point in a hyperbola he one diagonal of a parallelogram, two of whose sides coincide with the asymptotes, the other diagonal is parallel to the tangent at the point. 41.] Let us reverse this question, and seek the envelope of a line which cuts off from two fixed axes a triangle of constant area. If the axes be in the directions of a and fi, the intercepts may evidently be written at and y . Hence the equation of the line is (§30) p = at-\-x{Y—aty The condition of envelopment is, obviously, (see Chap. IX.) dp = 0. This gives =\a-x{^ + a)\dt+ {^-at)dx*. Hence {\—x)dt — tdx=0, J X ^, dx ^ and — — dt+ -^ = 0. * We are not here to equate to zero the coefficients of dt and dx; for we must remember that this equation is of the form =pa + q$, where p and q are numbers ; and that, so long as a and are actual and non-parallel vectors, the existence of such an equation requires i> = 0, 5 = 0. 43- J VECTORS, AND THEIK COMPOSITION. 21 From these, at once, x = \, since dx and dt are indeterminate. Thus the equation of the envelope is the hyperbola as before ; a, ;3 being portions of its asymptotes. 42.] It may assist the student to a thorough comprehension of the above process, if we put it in a slightly different form. Thus the equation of the enveloping line may be written p = ai!(l-a!) + /3*, which gives dp = = ad {t (1 —x))+^d (-) • Hence, as a is not parallel to /3, we must have d{t{l-x)) = (i, ^(f) = 0; and these are, when expanded, the equations we obtained in the preceding section. 43.] For farther illustration we give a solution not directly em- ploying the differential calculus. The equations of any two of the enveloping lines are p = at + X 0-: at\t p =aifi + a?i(— -ai5i)> t and <i being given, while x and ajj are indeterminate. At the point of intersection of these lines we have (§ 26), t{l-x) = ^i(l-«i), \ X _Xi > These give, by eliminating x^, t{\-x) = ti{l-^x), t or X =■ . • ti + t Hence the vector of the point of intersection is 22 QUATERNIONS. [44. and thus, for the ultimate intersections, where ^^ = 1, p = ^ (a^ + y) as before. CoE. (1). If tt^ = 1, a + /3 . or the intersection lies in the diagonal of the parallelogram on a, j3. Cob. (2). If ti = mt, where m is constant, mta + — V P = ■ But we have also iv = m+ 1 1 i+l Hence tAe locus of a point which divides in a given ratio a line cutting off a given area from, two fixed axes, is a hyperbola of which these axes are the asymptotes. Cor. (3). If we take tt^ (^+ ^1) = constant the locus is a parabola ; and so on. 44.] The reader who is fond of Anharmonic Ratios and Trans- versals will find in the early chapters of Hamilton's Elements of Quaternions an admirable application of the composition of vectors to these subjects. The Theory of Geometrical Nets, in a plane, and in space, is there very fully developed ; and the method is shewn to include, as particular cases, the processes of Grassmann's Ausdehnungslehre and Mobius' Barycentrische Calcul. Some very curious investigations connected with curves and surfaces of the second and third orders are also there founded upon the composition of vectors. EXAMPLES TO CHAPTER I. 1. The lines which join, towards the same parts, the extremities of two equal and parallel lines are themselves equal and parallel. {Euclid, I. xxxiii.) 2. Find the vector of the middle point of the line which joins EXAMPLES TO CHAPTER I. 23 the middle poiats of the diagonals of any quadrilateral, plane or gauche^ the vectors of the corners being given ; and so prove that this point is the mean point of the quadrilateral. If two opposite sides be divided proportionally, and two new quadrilaterals be formed by joining the points of division, the mean points of the three quadrilaterals lie in a straight line. Shew that the mean point may also be found by bisecting the line joining the middle points of a pair of opposite sides. 3. Verify that the property of the coefficients of three vectors whose extremities are in a line (§ 30) is not interfered with by altering the origin. 4. If two triangles ABC, abc, be so situated in space that Aa, Bb, Cc meet in a point, the intersections of AB, ah, of BG, be, and of CA, ca, lie in a straight line. 5. Prove the converse of 4, i. e. if lines be drawn, one in each of two planes, from any three points in the straight line in which these planes meet, the two triangles thus formed are sections of a common pyramid. 6. If five quadrilaterals be formed by omitting in succession each of the sides of any pentagon, the lines bisecting the diagonals of these quadrilaterals meet in a point. (H. Fox Talbot.) 7. Assuming, as. in § 7, that the operator cos 6 + \/— 1 sin 6 turns any radius of a given circle through an angle 6 in the positive direction of rotation, without altering its length, deduce the ordinary formulae for cos [A + B), cos {A—B), sin {A + B), and sin [A—B), in terms of sines and cosines of A and B. 8. If two tangents be drawn to a hyperbola, the line joining the centre with their point of intersection bisects the lines joining the points where the tangents meet the asymptotes : and the tangent at the point where it meets the curves bisects the intercepts of the asymptotes. 9. Any two tangents, limited by the asymptotes, divide each other proportionally. 10. If a chord of a hyperbola be one diagonal of a parallelogram whose sides are parallel to the asymptotes, the other diagonal passes through the centre. 11. Shewthat p = x^ a + f ^-\-{x-\-yf y is the equation of a cone of the second degree, and that its section by the plane _ pa + g^ + ry ^~ p + q+r 24 QUATERNIONS. is an ellipse which touches, at their middle points, the sides of the triangle of whose corners a, /3, y are the vectors. (Hamilton, Elements, p. 96.) 12. The lines which divide, proportionally, the pairs of opposite sides of a gauche quadrilateral, are the generating lines of a hyper- bolic paraboloid. (Ibid. p. 97.) 13. Shew that p = x^a + y^fi + z^y, where x + y + z = 0, represents a cone of the third order, and that its section by the plane '' ~ p + q + r is a cubic curve, of which the lines P + 1 are the asymptotes and the three (real) tangents of inflexion. Also that the mean point of the triangle formed by these lines is a conjugate point of the curve. Hence that the vector a-f-(3 + y is a conjugate ray of the cone. (Ibid. p. 96.) CHAPTER ir. PRODUCTS AND QUOTIENTS OF VECTOES. 45.] We now come to the consideration of points in which the Calculus of Quaternions differs entirely from any previous mathe- matical method ; and here we shall get an idea of what a Qua- ternion is, and whence it derives its name. These points are fundamentally, involved in the novel use of the symbols of mul- tiplication and division. And the simplest introduction to the subject seems to be the consideration of the quotient, or ratio, of two vectors. 46.] If the given vectors be parallel to each other, we have already seen (§ 22) that either may be expressed as a numerical multiple of the other; the multiplier being simply the ratio of their lengths, taken positively if they are similarly directed, nega- tively if they run opposite ways. 47.] If they be not parallel, let OA and OB be drawn parallel and equal to them from any point ; and the question is reduced to finding the value of the ratio of two vectors drawn from the same point. Let us try to find upon how many distinct numbers this ratio depends. We may suppose OA to be changed into 0£ by the following processes. 1st. Increase or diminish the length of OA till it becomes equal to that of OB. For this only one number is required, viz. the ratio of the lengths of the two vectors. As Hamilton remarks, this is a positive, or rather a signless, number. 2nd. Turn OA about until its direction coincides with that of OB, and (remembering the effect of the first operation) 26 QUATERNIONS. [48. we see that the two vectors now coincide or become identical. To specify this operation three more numbers are required, viz. two angles (such as node and inclina- tion in the case of a planeVs orbit) to fix the plane in which the rotation takes place, and one angle for the amount of this rotation. Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends upon four distinct numbers, whence the name quaternion. The particular case of perpendicularity of the two vectors, where their quotient is a vector perpendicular to their plane, is fully con- sidered below ; §§ 64, 65, 72, &c. 48.] It is obvious that the operations just described may be performed, with the same result, in the opposite order, being per- fectly independent of each other. Thus it appears that a quaternion, considered as the factor or agent which changes one definite vector into another, may itself be decompofed into two factors of which the order is immaterial. The stretching factor, or that which performs the first operation in § 47, is called the Tensou, and is denoted by prefixing T to the quaternion considered. The turninff factor, or that corresponding to the second operation in § 47, is called the Versoe, and is denoted by the letter U prefixed to the quaternion. 49.] Thus, if OA = a, OB = fi, and if q be the quaternion which changes a to /3, we have 13 = qa, which we may write in the form — = q, or ^a-i = q, a if we agree to defne that — .a = |3a-i. o = p. Here it is to be particularly noticed that we write q before a to signify that a is multiplied by q, not q multiplied by a. This remark is of extreme importance in quaternions, for, as we shall soon see, the Commutative Law does not generally apply to the factors of a product. We have also, by §§ 47, 48, q=TqUq=UqTq, 51.] PRODUCTS AND QUOTIENTS OF VECTORS. 27 where, as before, Tq^ depends merely on the relative lengths of a and j3, and Uq depends solely on their directions. Thus, if Oi and jSj be vectors of unit length parallel to a and j3 respectively, ^^^^^ u^^- = U^. As will soon be shewn, when a is perpendicular to ^, the versor of the quotient is quadrantal, i. e. it is a unit-vector. 50.] We must now carefully notice that the quaternion which is the quotient when /3 is divided by a in no way depends upon i\^e- absolute lengths, or directions, of these vectors. Its value will remain unchanged if we substitute for them any other pair of vectors which (1) have their lengths in the same ratio, (2) have their common plane the same or parallel, and (3) make the same angle with each other. Thus in the annexed figure 6^1^ OB 0^1 " OA if, and only if, ^^^ O.Aj^ OA (2) plane AOS parallel to plane A^O^B^, (3) I.AOB = LA^O^B^. [Equality of angles is understood to include similarity in direction. Thus the rotation about an upward axis is negative (or right-handed) from OA to OB, and also from Oj A^ to 0^ B^r\ 51.1 The Reciprocal of a quaternion q is defined by the equation, 1 -1 1 q- = qi ^=1. Hence if - = ?> ov a ^=qa, a 1 J we must have 'B~a ~ ^ ' a ■, For this gives -.p = q '^.qa, and each member of the equation is evidently equal to a. ^^/A 28 QUATERNIONS. [52. Or, we may reason thus, q changes 61 to 0£, q-^ must therefore change OB to OA, and is therefore expressed by - (§ 49). The tensor of the reciprocal of a quaternion is therefore the reciprocal of the tensor ; and the versor differs merely by the reversal of its representative angle. The versor, it must be remembered, gives the plane and angle of the turning — ^it has nothing to do with the extension. 52.] The Conjugate of a quaternion q, written Kq, has the same tensor, plane, and angle, only the angle is taken the reverse way. Thus, if OA, OB, OA', lie in one plane, and if 0A'= OA, and LA:0B=IA0B, we have OB ,6b ■ . n TT -z=^ = a, and-^=- = coniugate 01 q ■=■ Kq. OA ^' OA' ^^ ^ ^ By last section we see that Kq = {Tqfq-\ Hence qKq = Kq.q = {Tqf. This proposition is obvious, if we recollect that the " tensors of q and Kq are equal, and -that the versors are such that either annuls the effect of the other. The joint effect of these factors is therefore merely to multiply twice over by the common tensor. 53.] It is evident from the results of § 50 that, if a and ^ be of equal length, their quaternion quotient becomes a versor (the tensor being unity) and may be represented indifferently by any one of an infinite number of ares of given length lying on the circumference of a circle, of which the two vectors are radii. This js of considerable importance in the proofs which follow. fyo Thus the versor ^=^ may be represented in magnitude, plane, and direction (§ 50) by the arc AB, which may in this extended sense be written AB. And, similarly, the versor ' is repre- sented by A^B^ which is equal to (and measured in the same direction as) AB if jLAiOBi = LAOB, i.e. if the versors are equal, in the quaternion meaning of the word. 56. j PRODUCTS AKD QTJOTIEN.TS OF VECTORS. 29 54.] By the aid of this process,''Vhen a versor is represented as an arc of a great circle on the unit-sphere, we can easily prove that qiiaternion multiplication is not generally commutative. Thus let q be the versor j!b or ^=- • ^ ^ OA Make BC = AB, (which, it must be remembered, makes the points A, B, C lie in one great circle), then q^ may also be represented by • ^ OB In the same way any other versor r may be represented by BB or BB and bv -=- or -^=- • ^ OB OB The line OB in the figure is definite, and is given by the inter- section of the planes of the two versors ; being the centre of the unit-sphere. Now rOB = OB, and qOB = 00, Hence _ qrOB=6c, 00 '-^ or qr = -=■ > and may therefore be represented by the arc BC of a great circle. But rq is easily seen to be represented by the arc AB. For q02 = OB, and rOB = OB, — OB whence rq OA = OB, and rq = -=- • OA Thus the versors rq and qr, though represented by arcs of equal length, are not generally in the same plane and are therefore un- equal : unless the planes of q and r coincide. Calling OA a, we see that we have assumed, or defined, in the above proof, that q.ra = qr.a and r.qa = r'q.a when qa, ra, q.ra, and r.qa are all vectors. 55.J Obviously CB is Kq, BB is Kr, and CB is K(qr). But CB = BB.CB, which gives us the very important theorem K{qr) =Kr.Kq, i.e. the conjugate of tM product of two quaternions is the product of their conjugates in inverted order. 56.] The propositions just proved are, of course, true of quater- nions as well as of versors ; for the former involve only an additional 30 QUATERNIONS. [57- numerical factor which has reference to the length merely, and not the direction, of a vector (§48). 57.] Seeing thus that the commutative law does not in general hold in the multiplication of quaternions, let us enquire whether the Associative Law holds. That is, itj), q, r be three quaternions, have we jtj.g'r = j)q.r ? This is, of course, obviously true if jo, q, r be numerical quantities, or even any of the imaginaries of algebra. But it cannot be con- sidered as a truism for symbols which do not in general give M = iP- 58.] In the first place we remark that ^, q, and r may be con- sidered as versors only, and therefore represented by arcs of great . circles, for their tensors may obviously (§ 48) be divided out from both sides, being commutative with the versors. Let AB =p, BB = CA = q, aadi IE =z r. Join BC and produce the great circle till it meets EF in H, and make KH = FE = r, Bxidi EG = GB = pq (§ 54). Join GK. Then KG=HG.n]: = pq.r. Join FB and produce it to meet AB in M. Make lM=fb, and MN=AB, ~^frlf^ and join NL. Then LN= MN.£M = p.qr. Hence to shew that p.qr = pq.r all that is requisite is to prove that LN, and KG, described as above, are equal a/rcs of the same great circle, since, by the figure, they are evidently measured in the same direction. This is perhaps most easily efiected by the help of the fundamental properties of the curves known as Sjokerical Conies. As they are not usually familiar to students, we make a slight digression for the purpose of proving these fundamental properties ; after Chasles, by whom and Magnus they were discovered. An independent proof of the asso- ciative principle will presently be indicated, and in Chapter VII we shall employ quaternions to give an independent proof of the theorems now to be established. 59.*] Dbf. a spherical conic is the curve of intersection of a cone of the second degree with a sphere, the vertex of the cone being the centre of the sphere. 59-J PRODUCTS AND QUOTIENTS OF VKCTORS. 31 Lemma. If a cone have one silHes of circular sections, it bas another series, and any two circles belonging to different series lie on a sphere. This is easily proved as follows. Describe a sphere, A, cutting the cone in one circular section, C, and in any other point whatever, and let the side OpP of the cone meet A in p, P; P being a point in C. Then PO-Op is constant, and, therefore, since P lies in a plane, p lies on a sphere, a, passing through 0. Hence the locus, c, of js is a circle, being the intersection of the two spheres A and a. Let OqQ be any other side of the cone, q and Q being points in c, C respectively. Then the quadrilateral qQPp is inscribed in a circle (that in which its plane cuts the sphere, J) and the exterior angle at p is equal to the interior angle at Q. If OL, OMhe the lines in which the plane POQ cuts the cyclic planes (planes through parallel to the two series of circular sections) they are obviously parallel to pq, QP, respectively ; and therefore LLOp = LOpq = loqp = iMoq. Let any third side, OrE, of the cone be drawn, and let the plane OPR cut the cyclic planes in 01, Om respectively. Then, evidently, L10L= L qpr, Z.MOm= LQPR, and these angles are independent of the position of the points p and P, if Q and R be fixed points. In a section of thp above diagram by a sphere whose centre is 0, IL, Mm are the great circles which repre- sent the cyclic planes, PQ,R is the spherical conic which represents the cone. The point P represents the line OpP, and so with the others. The propositions above may now be stated thus Arc PL = arc MQ ; and, if Q and R be fixed, Mm and IL are constant arcs whatever be the position of P. 32 QUATERNIONS. [60. 60.] The application to § 58 is now obvious. In the figure of that article we have FE=ES, EI) = CA, Sg = CB, LM = FI). Hence L, C, G, D are points of a spherical conic whose cyclic planes are those of AJB, FE. Hence also KG passes through L, and with LM intercepts on AB an arc equal to AB. That is, it passes through N, or KG and LN are arcs of the same great circle : and they are equal, for G and L are points in the spherical conic. Also, the associative principle holds for any number of quaternion factors. For, obviously, qr.st = qrs.t = &c., &c., since we may consider qr as a single quaternion, and the above proof applies directly. 61.] That quaternion addition, and therefore also subtraction, is commutative, it is easy to shew. For if the planes of two quaternions, q and r, intersect in the line OA, we may take any vector OA in that line, and at once find two others, OB and OC, such that 0B= qOA, and OC=r OA. And {q + r)0A=0B+0C=0C+0B={r+q)6A, since vector addition is commutative (§ 27). Here it is obvious that {q + r)OA, being the diagonal of the parallelogram on OB, OC, divides the angle between OB and OC in a ratio depending solely on the ratio of the lengths of these lines, i. e. on the ratio of the tensors of q and r. This will be useful to us in the proof of the distributive law, to which we proceed. 62.] Quaternion multi- JD plication, and therefore di- vision, is distributive. One simple proof of this depends on the possibility, shortly to be proved, of representing an^ quaternion as a linear function of three given rect- angular unit-vectors. And when the proposition is thus established, the associative principle may readily be deduced froin it. But we may employ for its proof the properties of Spherical 63-] PfiODUCTS AND QUOTIENTS OP VECTORS. 33 Conies already employed in demAsttating the truth of the asso- ciative principle. For continuity we give an outline of the proof by this process. Let BA, GA represent the versors of q and r, and be the great circle whose plane is that of js. Then, if we take as operand the vector OA, it is obvious that U{q + r) will be represented by some such arc as BA where B, B, C are in one great circle ; for {q + r) OA is in the same plane as q OA and rOA, and the relative magnitudes of the arcs BB and BC depend solely on the tensors of q and r. Produce BA, BA, CA to meet be in b, d, e respectively^ and make ^ M = BA, m= BA, Gc= CA. Also make b^ = dh = cy =^. Then E, F, G, A lie on a spherical conic of which BG and be are the cyclic arcs. And, because bfi = dh = cy, pE, hF, y G, when produced, meet in a point R which is also on the spherical conic (§ 59*). Let these arcs meet BG in J, B, K respectively. Then we have JHz= E^ = pUq, LH=M =pU{q + r), KE= Gy =p Ur. Also fj= BB, and EL = CB. And, on comparing the portions of the figure bounded respectively by HKJ and by AGB we see that (when considered with reference to their effects as factors multiplying OH and OA respectively) J) U(^qjf.r) bears the same relation to jo Uq and jo Ur that lf{q + r)\)G2iXsto Uq&xA Ur. But T{q + r)U{q + r) = q + r = TqUq + TrUr. Hence T^ + r).jpU{q + r) = Tq.p Uq + Tr.p Ur ; or, since the tensors are mere numbers and commutative with all other factors, p{q + r) = pq +pr. In a similar manner it may be proved that {q + r)p = qp + rp. And then it follows at once that (p + q) (r+s) =pr +ps + qr-j-qs. 63.1 By similar processes to those of § 53 we see that versors, and therefore also quaternions, are subject to the index-law q'^.q" = j'"+", at least so long as m and n are positive integers. D 34 QUATERNIONS. [64. The extension of this property to negative and fractional ex- ponents must be deferred until we have defined a negative or fractional power of a quaternion. 64.] We now proceed to the special case of guadrantal versors, from whose properties it is easy to deduce all the foregoing results of this chapter. These properties were indeed those whose in- vention by Hamilton in 1843 led almost intuitively to the esta- blishment of the Quaternion Calculus. We shall content ourselves at present with an assumption, which will be shewn to lead to consistent results-; but at the end of the chapter we shall shew that no other assumption is possible, following for this purpose a very curious quasi-metaphysical speculation of Hamilton. 65.] Suppose we have a system of three mutually perpendicular unit- vectors, drawn from one point, which we may call for short- ness I, J, K. Suppose also that these are so situated that a positive (i. e. left-handed) rotation through a right angle about / as an axis brings J to coincide with K. Then it is obvious that positive quadrantal rotation about / will make K coincide with /; and, about K, will make I coincide with /. For definiteness we may suppose J to be drawn eastwards, J north- wards, and K upwards. Then it is obvious that a positive (left- handed) rotation about the eastward line (7) brings the northward line (i7) into a vertically upward position {K); and so of the others. 66.] Now the operator which turns J into Z" is a quadrantal versor (§ 53) ; and, as its axis is the vector I, we may call it i. Thus T'^^' °^ K=iJ- (1) Similarly we may put -= =j, or I=.jK, (2) and -Y = k, or J = hi. (3) [It may here be noticed, merely to shew the symmetry of the system we are explaining, that if the three mutually perpendicular vectors /, /, Xbe made to revolve about a line equally inclined to all, so that / is brought to coincide with J, J will then coincide with K, and X with I: and the above equations will still hold good, only (1) will become (2), (2) will become (3), and (3) will become (I)-] 67.] By the results of § 50 we see that -/_ K K ~ J' 69.] PEODUCTS AND QUOTIENTS OP VEOTOES. 35 1. e. a southward unit- vector bears the same ratio to an upward unit-vector that the latter does to a northward one; and therefore we have ~" -J=iK. — K Similarly -^^ = and -/ K -E -I J t, or or = ff, or -K=jl; -I=kJ. ■(4) .(5) (6) (7) 68.] By (4) and (1) we have. -J =iK= i{iJ) = i^J. Hence p = _ 1 And, in the same way, (5) and (2) give -^'=-1' (8) and (6) and (3) F ^ — 1 (9^ Thus, as the directions of /, J, K are perfectly arbitrary, we see that Ue square of every quad/rantal versor is negative unity. Though the following proof is in principle exactly the same as the foregoing, it may perhaps be of use to the student, in shewing him precisely the nature as well as the simplicity of the step we have taken. Let ABA' be a semicircle, whose centre is 0, and let OB be perpendicular to AOA'. Then ^=:^ , = q suppose, is a quadrantal OA OA' versor, and is evidently equal to -:=r; §§ 50, 53. _^ _ ^^ OA' OB 61' A' -ot Hence r = -=- OB OA OA 69.] Having thus found that the squares of i, J, h are each equal to negative unity ; it only remains that we find the values of their products two and two. For, as we shall see, the result is such as to shew that the value of any other combination whatever of i, j, Jc (as factors of a product) may be deduced from the values of these squares and products. Now it is obvious that _Z -I~ K D a _I__ . 36 QUATERNIONS. [70. (i. e. the versor which turns a westward unit- vector into an upward one will turn the upward into an eastward unit) ; or K = J{-I)=-jI* (10) Now let us operate on the two equal vectors in (10) by the same versor, i, and we have iK = i {—jl) = —ijl. But by (4) and (3) iK = -J =-kI. Comparing these equations, we have —ijl=-kl; or, by § 54 (end), ij = k,^ and symmetry gives jh = i, \ (11) hi = j. J The meaning of these important equations is very simple ; and is, in fact, obvious from our construction in § 54 for the multiplication of versors; as we see by the annexed figure, where we must re- member that i, j, ^^re quadrantal versor s whose planes are at right angles, so that the figure represents a hemisphere divided into quadrantal tri- angles. Thus, to shew that ij = k, we have, being the centre of the sphere, N, E, S, W the north, east, south, and west, and ^the zenith (as in § 65) ; j6W= 6z, whence ijOW=^iOZ= OS - kOW. 70.] But, by the same figure, i_ON=OZ, _ _ _ whence jiON = jOZ= OE = -OW=-kON. 71. J From this it appears that ji=-k, \ and similarly kj =- — i, > (12) ik = -J, ) and thus, by comparing (11), (/ = -i* = ^> ) jk=~kj=iA ((11), (12)). ki = —ik = J. ) * The negative sign, being a mere numerical &ctor, is evidently commutative with j ; indeed we may, if necessary, easily assure ourselves of the fact that to turn the negative (or reverse) of a vector through a right (or indeed any) angle, is the same thmg ae to turn the vector through that angle and then reverse it. 74-] PRODUCTS ANB QUOTIENTS OF VECTOBS. 37 These equations, along with i2=/=F=-l ((7), (8), (9)), contain essentially the whole of Quaternions. But it is easy to see that, for the first group, we may substitute the single equation V^=-l, (13) since from it, by the help of the values of the squares of i, J, h, all the other expressions may be deduced. We may consider it proved in this way, or deduce it afresh from the figare above, thus hON= 6W, jkON= j6W= 6Z, ijhON= ijOW=^i6Z= 68 = -0N. 72.] One most important step remains to be made, to wit the assumption -referred to in § 64. We have treated i,j, k simply as quadrantal versors ; and /, J, K as unit-vectors at right angles to each other, and coinciding with the axes of rotation of these versors. But if we collate and compare the equations just proved We have = .k, (11) .iJ= K, (1) \Ji=-k, (12) Ijl = -K, (10) with the other similar groups symmetrically derived from them. Now the meanings we have assigned to i, _;', k are quite inde- pendent of, and not inconsistent with, those assigned to I, J, Ki And it is superfluous to use two sets of characters when one will suffice. Hence it appears that «', /, k may be substituted for J, /, K; in other words, a unit-vector when employed as a factor may be con^ sidered as a quadrantal versor whose plane is perpendicular to the vector. This is one of the main elements of the singular simplicity of the quaternion calculus. 73.] Thus the product, and therefore the quotient, of two perpen- dicular vectors is a third vector perpendicular to hoth. Hence the reciprocal (§ 51) of a vector is a vector which has the opposite direction to that of the vector, and its length is the re- ciprocal of the length of the vector. The conjugate (§ 52) of a vector is simply the vector reversed. Hence, by § 52, if a be a vector {Taf = aKa = a (-a) = -a". 74.J We may now see that every versor may be represented by a power of a unit-vector. 38 QUATEENIONS. [75- For, if a be any vector perpendicular to i (which is an^ definite unit-vector), »a, = /3, is a vector equal in length to a, but perpendicular to both i and a ; i^a = — a, i^a =—ia = — /3, i*a = — i/3 = —i^ a = a. Thus, by successive applications of i, a is turned round i as an axis through successive right angles. Hence it is natural to define i*" as a versor which turns any vector perpendicular to i through m right angles in the positive direction of rotation about i as an axis. Here m may have any real value whatever, whole or fractional, for it is easily seen that analogy leads us to interpret a negative value of m as corresponding to rotation in the negative direction. 75.] From this again it follows that any quaternion may be expressed as a power of a vector. For the tensor and versor elements of the vector may be so chosen that, when raised to the same power, the one may be the tensor and the other the versor of the given quaternion. The vector must be, of course, perpendicular to the plane of the quaternion. 76.] And we now see, as an immediate result of the last two sections, that the index-law holds with regard to powers of a quaternion (§ 63). 77.] So far as we have yet considered it, a quaternion has been regarded as the product of a tensor and a versor : we are now to consider it as a sum. The easiest method of so analysing it seems to be the following. Let ^=- represent any quaternion. Draw BC perpendicular to OA, produced if neces- sary. Then, §19, OB = OC+CB. But, § 22, OC = xOA, where a; is a number, whose sign is the same as that of the cosine of Z AOB. Also, § 73, since CB is perpendicular to OA, CB = yOA, where y is a vector perpendicular to OA and CB, i.e. to the plane of the quaternion. TT OB (vOl + yOA Hence -^^^ = =i — =a! + v. OA OA 79-] PRODUCTS AND QUOTIENTS OF VECTORS. 39 Thus a quaternion, in general, mSy be decomposed into the sum of two parts, one numerical^ the other a vector. Hamilton calls them the SCALAE, and the vector, and denotes them respectively by the letters S and T prefixed to the expression for the quaternion. 78.] Hence q = Sq+ Vq, and if in the above example OB M=^ then OB = 0C+ CB = Sq.OA-\- Fq.Ol*. The equation above gives OC^Sq.OA, GB = rq.OA. 79.] If, in the figure of last section, we produce BG to B, so as to double its length, and join OB, we have, by § 52, ^=Kq = SKq^YKq; .-. 6B=0C + CB = 8Kq.62+rKq.0J. Hence OC = SKq.OA, and CB = rKq.OA. Comparing this value of OC with that in last section, we find 8Kq = 8q, (1) or the scalar of the conjugate of a quaternion is equal to the scalar of the quaternion. Again, CB — — CB by the figure, and the substitution of their values gives VKq^-Vq, '. (2) or the vector of the conjugate of a quaternion is the vector of the quaternion reversed. We may remark that the results of this section are simple con- sequences of the fact that the symbols S, V, K are commutative f. ^ Thus SKq = K8q = Sq, since the conjugate of a number is the number itself; and VKq = KVq = -rq{\ 73). * The points are inserted to shew that S and Y apply only to q, and not to qOA . + It is curious to compare the properties of these quaternion symbols with those of the Elective Symbols of Logic, as given in Boole's wonderful treatise on the LoAm of Thought; and to think that the same grand science of mathematical analysis, by processes remarkably similar to each other, reveals to ns truths in the science of position far beyond the powers of the geometer, and truths of deductive reasoning to which unaided thoug'ht could never have led the logician. 40 QUATERNIONS. [8o. Again, it is obvious that :^Sq = S2q, l,Fq= Flq, and thence SKq = Kl,q. 80.] Since any vector whatever may be represented by xi + yj-i-zl! where x, y, z are numbers (or Scalars), and i, j, h may be any three non-coplanar vectors, §§ 23, 25 — though they are usually under- stood as representing a rectangular system of unit-vectors — and since any scalar may be denoted by w; we may write, for any quaternion q, the expression q = w-\-m-\-yj->rzh (§ 78). Here we have the essential dependence on four distinct numbers, from which the quaternion derives its name, exhibited in the most simple form. And now we see at once that an equation such as where §^= v/+x'i-\-i^j-\-/h, involves, of course, the ybwr equations vf=w, af= X, y'=y, i^—z. 81.] "We proceed to indicate another mode of proof of the dis- tributive law of multiplication. We have already defined, or assumed (§61), that - + - = > a a a or ^a-i+ya-i = (^ + y)a-i, and have thus been able to understand what is meant by adding two quaternions. But, writing o for a~^, we see that this involves the equality [fi + y)a = /3a + ya; from which, by taking the conjugates of both sides, we derive And a combination of these results (putting /3 + y for a' in the latter, for instance) gives (^ + y)(^+/) = (/3+y)^+G3 + y)/ = i3/3'-|- yi3'+ /3y'+ yy by the former. Hence the distributive principle is true in the multiplication of vectors. It only remains to shew that it is true as to the scalar and 83.] PEODUCTS AND QUOTIENTS OF VECTOES. 41 vector parts of a quaternion, and then we shall easily attain the general proof. Now, if a be any scalar, a any vector, and q^ any quaternion, (a + aj 3' = «§' + aq. For, if ;3 be the vector in which the plane of §' is intersected by a plane perpendicular to a, we can find other two vectors, y and 8, in these planes such that And,. of course, a may be written ■—; so that ^ ^^ y3 8 8 -'^6 +6-'^8 + ^ 6 = aq-\-aq^. And the conjugate may be written /(a' + o') = ?V + /a' (§55). Hence, generally, (a + a)(3 + ;8) =ffli + a/3 + Ja+a;8; or, breaking up « and 5 each into the sum of two scalars, and a, /3 each into the sum of two vectors, («i + aa + oi + Og) ( *! + «2 + ^1 + /Sg) = K + «2) (*i + ^2) + («i + «.) (/3i + /Sz) + {\ + ^2) («i + "2) + («l + «2)(^l + /32) (by what precedes, all the factors on the right are distributive, so that we may easily put it in the form) = («i + «i) (^1 + /5i) + K + «i) (*2 + /32) + («2 + a^) («i + /3i) + («2+a2)(52 + /32)- Putting ai + ai=j9, «2 + a2 = $', ^i + ZSj = /, 5a + /32=», we have ( J" + ?) 0" + *) =/'>" + i'»+$'»' + S'*. 83.] For variety, we shall now for a time forsake the geometrical mode of proof we have hitherto adopted, and deduce some of our next steps from the analytical expression for a quaternion given in § 80, and the properties of a rectangular system of unit-vectors as in § 71. We will commence by proving the result of § 77 anew. 83.] Let a = xi + yj-\-zh. 42 QUATERNIONS. [84. Then, because by § 7 1 every product or quotient of i, J, h is reducible to one of them or to a number, we aye entitled to assume ^ = ^ = a) + ^J + 77y+C^, where co, f, 77, f are numbers. This is the proposition of § 80. 84.] But it may be interesting to find to, £, tj, f in terms of x, y, z, af,^, z. We have ^ = qa, or x'i-\-y'j-\-^k = (a, +^»4- v'+ C^) {xi + yj+zk) as we easily see by the expressions for the powers and products of hji '^j given in § 71. But the student must pay particular attention to the order of the factors, else he is certain to make mistakes. This (§ 80) resolves itself into the four equations 0= ^SB + riy + Cm, x'=(x,x +i?«— fy, ^=wy—iz +Ca?, /= a)Z + $y—r]X. The three last equations give xx'+yy' + zz'= a) {x^ + y^ + z^), which determines m. Also we have, from the same three, by the help of the first, ix' + riy'+C/=0; which, combined with the first, gives ^ = ■n ^ C . y/—zy' zuZ—x/ x^—yx^' and the common value of these three fractions is then easily seen to be 1 x'^+y^+ z^ It is easy enough to interpret these expressions by means of ordinary coordinate geometry : but a much simpler process will be furnished by quaternions themselves iti the next chapter, and, in giving it, we shall refer back to this section. 85.] The associative law of multiplication is now to be proved by means of the distributive (§ 81). We leave the proof to the student. He has merely to multiply together the factors w + xi + yj + zk, w'+x'i + yj+Zk, and w" + x"i + y"j+/'k, as follows : — First, multiply the third factor by the second, and then multiply the product by the first ; next, multiply the second factor by the 88.J PRODUCTS AND QUOTIENTS OP VECTORS. 43 first and employ the product to multiply the third : always re- membering that the multiplier in any product is placed lefore the multiplicand. He will find the scalar parts and the coefficients of i,j, Te, in these products, respectively equal, each to each. 86.] With the same expressions for a, )3, as in section 83, we have a^ = {xi + yj + zh) {x'i + y'j + ^h) = - [xx' +yy' + zz') + ly/ -zy')i + {zx'-x^)j-\- {xy" -yx')k. But we have also ^a= — {xx'+ yy' +zz')- {y/ — z/) i - {zaf - xz')j — (a/ - yx) k The only difference is in the sign of the vector parts. Hence Sa^ = Spa, (1) ral3=-r^a, (2) also afi + ^a = 2Sa^, (3) al3-^a = 2rap, (4) and, finally, by § 79, a^ = K^a (5) 87.] If a = /3 we have of course (§ 25) x = x', y=y', z = z', and the formulae of last section become a^ — ^a = a' = —(x' + y^ + z') ; vsrhich was anticipated in § 73, where we proved the formula and alsOj to a certain extent, in § 25. 88.] Now let q and / be any quaternions, then S.qr = S.{Sq+ Vq) {Sr+ Fr), = S.{SqSr+Sr.Vq + Sq.Fr+ FqFr), = SqSr+SFqFr, since the two middle terms are vectors. Similarly, S.rq = SrSq + SFr Fq. Hence, since by (1) of § 86 we have SFqFr = SFrFq, we see that S.qr = S.rq, (1) a formula of considerable importance. It may easily be extended to any number of quaternions, because, r being arbitrary, we may put for it rs. Thus we have S.qrs = S.rsq, = S.sqr by a second application of the process. In words, we have the theorem — tAe scalar of ike product of any number of given quaternions depends only upon the cyclical order in which they are arranged. 44 QUATERNIONS. [89. 89.] An important case is that of three factor.s, each a vector. The formula then becomes S.a^y = S.^ya = S.ya^. But S. aPy = Sa{Spy + V^y) = SaVfiy, since aSfty is a vector, = -Saryp, by (2) of §86, = -Sa{SyP+ryP) = -S.ayfi. Hence tke scalar of the product of three vectors changes sign when the cyclical order is altered. Other curious propositions connected with this will be given later, as we wish to devote this chapter to the production of the fundamental formulae in as compact a form as possible. 90.] By (4) of §86, 2F^y=:/3y-yi8. Hence 2FaFfiy = Fa {fiy — y^) (by multiplying both by a, and taking the vector parts of each side) = F{apy+pay—pay—ayfi) (by introducing the null term ^ay—^ay). That is 2rar^y=r.{aP + l3a)y-F(fiSay + l3Fay + Say.p+Fay.p) = F{2Sa^)y-2F^Say (if we notice that F.Fay.^ =~ F^Fay, by (2) of § 86). Hence FaF^y = ySa^—fiSya, (1) a formula of constant occurrence. Adding aS^y to both sides we get another most valuable formula F.afiy = aSfiy-^Sya + ySa^; (2) and the form of this shews that we may interchange y and o without altering the right-hand member. This gives F.a^y = F.y^a, a formula which may be greatly extended. 91.] We have also FFa^Fyh = - FFyb Fafi by (2) of § 86 : = bSyFap-ySbFafi = bS.afiy-yS.a^b, = - pSa, Fyh + axS/3 Fyh = -pS. ayh + aS. j8y8, all of these being arrived at by the help of § 90 (1) and of § 89 ; and by treating alternately Fa/3 and Fyb as simple vectors. Equating two of these values, we have bS.al3y = aS.I3yb + l38.yab + yS.al3t, (3) 93-J PRODUCTS AND QUOTIENTS OF VECBOES. 45 a very useful formula, expressing any vector whatever in terms of three given vectors. 93.] That such an expression is possible we knew already by § 23. For variety we may seek another expression of a similar character^ by a process which differs entirely from that employed in last section. a, /3, y being any three vectors, we may derive from them three others Faj3, V^y, Vya ; and, as these will not generally be coplanar, any other vector 6 may be expressed as the sum of the three, each multiplied by some scalar (§ 23). It is required to find this ex- pression for 5. Let h=go FajS + i/Vfiy + z Vya. Then 8yb = xS.yajS = xS.a^y, the terms in ^ and z going out, because Syri3y = S.yPy = S/Sy^ = y^SjS = 0, for y^ is (§ 73) a number. Similarly /S/38 = zS.^ya = zS.aj3y, and Sab = yS.a^y. Thus bS.apy = Fa^Syb + FjSySab + VyaS^b (4) 93.] We conclude the chapter by shewing (as promised in § 64) that the assumption that the product of two parallel vectors is a number, and the product of two perpendicular vectors a third vector perpendicular to both, is not only useful and convenient, but absolutely inevitable, if our system is to deal indifferently with all directions in space. We abridge Hamilton's reasoning. Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed ; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them ; for by changing the sign of both factors the product is unchanged, whereas, as the whole system has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be' wholly or partly a number, because on inverting one of them the sign of that number ought to change; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and (from the symmetry of space) ought to leave the number 46 QUATERNIONS. unchanged. Hence the product of two perpendicular vectors must be a vector, and a simple extension of the same reasoning shews that it must be perpendicular to each of the factors. It is easy to carry this farther, but enough has been said to shew the character of the reasoning. EXAMPLES TO CHAPTER II. 1 . It is obvious from the properties of polar triangles that any mode of representing versors by the sides of a triangle must have an equivalent statement in which they are represented by angles in the polar triangle. Shew directly that the product of two versors represented by two angles of a spherical triangle is a third versor represented by the supplement of the remaining angle of the triangle; and determine the rule which connects the directions in which these angles are to be measured. 2. Hence derive another proof that we have not generally m = IP- 3. Hence shew that the proof of the associative principle, § 57, may be made to depend upon the fact that if from any point of the sphere tangent arcs be drawn to a spherical conic, and also arcs to the foci, the inclination of either tangent arc to one of the focal arcs is equal to that of the other tangent arc to the other focal arc. 4. Prove the formulae 2S.apy = a^y—y^a, 2r.a/3y= a^y + y^a. 5. Shew that, whatever odd number of vectors be represented by a, 13, y, &c., we have always F.aPybe = V.eby^a, KajaybeCn = r.7jfe8y/3a, &c. 6. Shew that S.FajSFfiY^ya = -{S-ajSyf, r. ral3 r^y Vya = TajS {y^Sa^ - SfiySya) + , and F. ( Fa^ F. Ffiy Fya) = (fiSay- aSfiy) S.a^y. 7. If a, /3, y be any vectors at right angles to each other, shew that (a3 + ^3 + y 3) ;S.a^y = a* Fj3y + /3* Fya + y* Fafi. EXAMPLES TO CHAPTER 11. 47 8. If a, j3, y be non-eoplanar veifcors, find the relations among the six scalars, x, y, z and f, t\, f, which are implied in the equation xa->ry^-^zy = iFfSy + r\ Fya + QFa^. 9. If a, j3, y he any three non-eoplanar vectors, express any fourth vector, 6, as a linear function of each of the following sets of three derived vectors, r.yap, V.afiy, V.^ya, and V.ra^r^yVya, FV^yVyaVa^, F.FyaFapFfiy. 10. Eliminate p from the equations Sap = a, Sj3p = b, Syp = c, Sbp = d, where a, /3, y, 6 are vectors, and a, b, c, d scalars. 11. In any quadrilateral, plane or gauche, the sum of the squares of the- diagonals is double the sum of the squares of the lines joining the middle points of opposite sides. CHAPTER IIL INTERPKETATIONS AND TRANSFORMATIONS OF QUATERNION EXPRESSIONS. 94.] Among the most useful characteristics of the Calculus of Quaternions, the ease of interpreting its formulae geometrically, and the extraordinary variety of transformations of which the simplest expressions are susceptible, deserve a prominent place. We devote this Chapter to some of the more simple of these, to- gether with a few of somewhat more complex character but of constant occurrence in geometrical and physical investigations. Others will appear in every succeeding Chapter. It is here, perhaps, that the student is likely to feel most strongly the peculiar difficulties of the new Calculus. But on that very account he should endeavour to master them, for the variety of forms which any one formula may assume, though puzzling to the beginner, is of the most extraordinary advantage to the advanced student, not alone as aiding him in the solution of complex questions, but as affording an invaluable mental discipline. 95.] If we refer agiain to the figure of § 77 we see that 0C= OB cos AOB, CJB = OB sin JOB. Hence, if OJ = a, OB = p, and /.AOB = 6, we have OB = Tl3, OA = Ta, OC = Tfi cos d, CB = Tfi sin 0. o/3 OC Tl3 Hence S- = -^-r = -y==— cos5. a OA la Similarly rr^ = ^ = ^sin.. 9 7- J INTERPRETATIONS AND TRANSFORMATIONS. 49 Hence, if e be a unit- vector perpenoicular to o and /3, or UOA o. we have F- = -^ sin 0.{. a Ta 96.] In the same way we may shew that TVa^ = Ta Tfi sin 6, and Fa^ = Ta Tfi sin O.r, where 77= Urap = UF^- a Thus tAe scalar of the product of two vectors is the continued product of their tensors and of the cosine of the sitpplement of the contained angle. The tensor of the vector of the product of two vectors is the con- tinued product of their tensors and the sine of the contained angle ; and the versor of the same is a unit-vector perpendicular to both, and such that the rotation about it from the first vector (i. e. the multiplier') to the second is left-handed or positive. Hence TFa^ is doMe the area of the triangle two of whose sides are a, /3. 97.] (a.) In any triangle A£C we have AC = A£ + W. Hence IC^ = SAC AC = S.AC{AB + SC). With the usual notation for a plane triangle the interpretation of this formxila is —b^ = —be cos A— ab cos, C, or b= a cos C+c cos A (b.) Again we have, obviously, rABAC= FAS{A£ + BC) = FABBG, or cb sin A = ca sin B, sin A sin B sin C whence = — j— = • a c These are truths, but not truisms, as we might have been led to fancy from the excessive simplicity of the process employed. E 50 QUATERNIONS. [98. 98.] From § 96 it follows that, if a and /3 be both actual (i. e. real and non-evanescent) vectors, the equation Sa^ = shews that cos 6 = 0, or that a is perpendicular to /3. And, in fact, we know already that the product of two perpendicular vectors is a vector. Again, if ^„^ = 0, we must have sin ^ = 0, or a is parallel to /3. We know already that the product of two parallel vectors is a scalar. Hence we see that Sa^ = is equivalent to o = Fy/3, where y is an undetermined vector ; and that is equivalent to a = a;/3, where aj is an undetermined scalar. 99.] If we write, as in § 83, o — ix +ji/ + kz, /3 = ix'+j/ + M, we have, at once, by § 86, Sa^ = —xaf—yy'—z/ , ^x af , y 1/ z z' \ = -r/( - + I-£^ + — ) \ r r r r r r ' where r = -s/as^+j^^ +«^ /= \/«'^+y^+/^. Also r^ = ^|i:!:^i+ff:^-y+at.er,j. These express in Cartesian coordinates the propositions we have just proved. In commencing the subject it may perhaps assist the student to see these more familiar forms for the quaternion expressions ; and he will doubtless be induced by their appearance to prosecute the subject, since he cannot fail even at this stage to see how much more simple the quaternion expressions are than those to which he has been accustomed. 100.] The expression S.a&y may be written S ( Fa/3) y, because the quaternion a.^y may be broken up into of which the first term is a vector. 102.] INTERPEETATIONS AND TEANSFOEMATIONS. 51 But, by §96, S ( ra/3) y = TaTl3 sin 9 Sriy. Here Tr) = 1, let (|) be the angle between ?j and y, then finally S.a^y =-TaT^Ty sin 5 cos (|). But as ?j is perpendicular to a and /3, Ty cos ^ is the length of the perpendicular from the extremity of y upon the plane of a, /3. And as the product of the other three factors is (§ 96) the area of the parallelogram two of whose sides are a, ^, we see that the mag- nitude otS.apy, independent of its sign, is i^e volume of the parallel- epiped of which three coordinate edges are a, fi, y; or six times the volume of the pyramid which has a, ^, y for edges. 101.] Hence the equation S.apy = 0, if we suppose a, /S, y to be actual vectors, shews either that sin e = 0, or cos(^ = 0, i. e. two of the three vectors are parallel, or all three are pArallel to one plane. This is consistent with previous results, for if y = ^j3 we have S.aPy=:pS.afi^ = Q; and, if y be coplanar with a, fi, we have y =pa + qP, and S.al3y = S.al3{pa + ql3) = 0. 102.] This property of the expression S.a^y prepares us to find that it is a determinant. And, in fact, if we take a, ;3 as in § 83, and in addition ^ ^ ^^" +_^y ^ ^/'^ we have at once S.apy = —x" [yi^-zy')-f {zx'—xz) ^z" {x/ —yx'), =.— X y z x' y' / of' f z" The deterrhinant changes sign if we make any two rows change places. This is the proposition we met with before (§ 89) in the form s^afiy = ^S.jSay = S.^ya, &e. If we take three new vectors ai = ix+j'a^+^a/', yi = iz+J/+M', we thus see that they are coplanar if o, ;3, y are so. That is, if iS.al3y = 0, then (S.Oj/Sjyi = 0. E -2 52 QUATERNIONS. [103. 103.] We have, by § 52, {Tqf = qKq = {Sq+ Fq) (Sq- fq) (§ 79), = lSqf-{rqf by algebra, = {SqY+{Trqf (§73). liq = aj8, we have Kq = fia, and the formula becomes a/3.;8a = a''^^ = {Sa^f-{Va^f. In Cartesian coordinates this is (a!''+/+02)(a^2+/2+/2) More generally we have (r(gr))2 = qrK{qr) = qrKrKq (§ 55) = {Tqf {Trf (§ 52). If we write q =.w ■\-a = w +ix +jy + kz, r = w' + l3 =. w'+iaf+Jy'+k/; this becomes = {wio'—xx'—^^'—z/f + {loixf + «/«; +^/—z/)'^ + {w/ + 'u/y+zx'—x/y + {10/ +w'z+x/—ya/)^, a formula of algebra due to Euler. 104.] We have, of course, by multiplication, (a+/3)2 = a^ + aj3 + ^a + P^ = a' + 2Sa^ + fi'' (§86 (3)), Translating into the usual notation of plane trigonometry, this becomes c^ =za^-2ah cos C+ b% the common formula. Again, r(a+/3) (a-/3) = - rai3+ T/Sa = -2 FaiS (§ 86 (2)). Taking tensors of both sides we have the theorem, the jparallelogram whose sides are parallel and equal to the diagonals of a given paral- lelogram, has double its area (§ 96). Also iS(a + /3)(a-/3) = a^-^^ and vanishes only when a^ = /3^, or Ta—T^\ that is, the diagonals of a parallelogram are at right angles to one another, when, and only when, it is a rhombus. Later it will be shewn that this contains a proof that the angle in a semicircle is a right angle. 105.] The expression p = a^a'^ obviously denotes a vector whose tensor is equal to that of /3. But we have S.^ap = 0, so that p is in the plane of o, ^. Also we have Sap = Sa^, I06.] INTERPRETATIONS AND TRANSFOEMATIONS. 53 • so that /3 and p make equal angles with a, evidently on opposite sides of it. Thus if a be the perpendicular to a reflecting surface and /3 the path of an incident ray, p will be the path of the re- flected ray. Another mode of obtaining these results is to expand the above expression, thus, § 90 (2), p = 2a-^Sa^-^^ SO that in the figure of § 77 we see that if OA = a, and OJB = ^, we have OJ) = p = a^a~''^. Or, agaiuj we may get the result at once by transforming the equation to U- = U-- 106.] For any three coplanar vectors the expression p = afiy is (§ 101) a vector. It is interesting to determine what this vector is. The reader will easily see that if a circle be described about the triangle, two of whose sides are (in order) a and /3, and if from the extremity of /3 a line parallel to y be drawn again cutting the circle, the vector joining the point of intersection with the origin of a is the direction of the vector afiy. For we may write it in the form a p = a^^fi-^y = -{T^fafi-^y = -{T^f -y, which shews that the versor (-A which turns j3 into a direction parallel to a, turns y into a direction parallel to p. And this ex- presses the long-known property of opposite angles of a quadri- lateral inscribed in a circle. Hence if a, ^, y be the sides of a triangle taken in order, the tangents to the circumscribing circle at the angles of the triangle are parallel respectively to a^y, Pya, and ya)3. Suppose two of these to be parallel, i. e. let a/3y = x^ya = as ay 13 (§ 90), since the expression is a vector. Hence Py = xyp, which requires either x=\, Fy^ = or y || /3, a case not contemplated in the problem ; or a; = -l, S^y = 0, 54 QUATERNIONS, [107. i. e. the triangle is right-angled. And geometry shews us at once that this is correct. Again, if the triangle be isosceles, the tangent at the vertex is parallel to the base. Here we have wfi = a^y, or (X!{a + y) = a{a + y)y; whence x = y'' = a?, or Ty = Ta, as required. As an elegant extension of this proposition the reader may prove that the vector of the continued product a^yS of the vector-sides of a quadrilateral inscribed in a sphere is parallel to the radius drawn to the corner {a, 8). 107.] To exemplify the variety of possible transformations even of simple expressions, we will take two cases which are of frequent occurrence in applications to geometry. Thus T{p-\-a) = T{s>-a), [which expresses that if 02 = a, 0A'= —a, and OP = p, we have AP = A'F, and thus that P is any point equidistant from two fixed points,] may be written (p + a)^ = {p—af, or p'^ + iSap + a^ = p^ — ^Sap + a^ {^101), whence Sap = 0. This may be changed to ap+ pa = 0, or ap + Kap = 0, SU^ = 0, a or finally, TFU^ = 1, all of which express properties of a plane. Again, Tp = Ta may be written T - = 1, ^ a'' ^ a'' (p + aY-28a{p + a) = 0, p= {p + a)-'^a{p+a), S{p + a){p—a) = 0, or finally, T.{p + a){p-a) = 2TVap. I09-] INTERPRETATIONS AND TRANSFORMATIONS. 55 • All of these express properties of a sphere. They will be in- terpreted when we come to geometrical applications. 108.] "We have seen in § 95 that a quaternion may be divided into its scalar and vector parts as follows : — a a a Ta where 9 is the angle between the directions of a and /3, and e= UF- a is the unit- vector perpendicular to the plane of a and /3 so situated that positive (i. e. left-handed) rotation about it turns a towards /3. Similarly we have (§ 96) 0/3 = Sa^ + Fa^ = TaT^{-cose + esin0), 6 and e having the same signification as before. 109.] Hence, considering the versor parts alone, we have U- = cos6 + t sin d. a Similarly U^ = cos (j) + e sincj} ; (j) being the positive angle between the directions of y and /3, and e the same vector as before, if a, /3, y be coplanar. Also we have U- = cos {d + <t)) + e sin {6 + <(>). But we have always -•- = -, and therefore |3 a a pa a or cos (<^ + 5) + e sin ((/)-}- 5) = (cos ^ -f e sin ^) (cos 5 + e sin 0) = cos (\) cos 5— sin (^ sin 9 + e (sin (pcos6 + cos ^ sin 6), from which we have at once the fundamental formulae for the cosine and sine of the sum of two arcs, by equating separately the scalar and vector parts of these quaternions. And we see, as an immediate consequence of the expressions abovcj that cos me + esmme = (cos -f e sin BJ" if m be a positive whole number. For the left-hand side is a versor which turns through the angle m5 at once, while the right-hand 56 QUATERNIONS. [lIO. side is a versor which effects the same object by m successive turn- ings each through an angle Q. See § 8. 110.] To extend this proposition to fractional indices we have only to write - for Q, when we obtain the results as in ordinary trigonometry. From De Moivre's Theorem, thus proved, we may of course deduce the rest of Analytical Trigonometry. And as we have already deduced, as interpretations of self-evident quaternion trans- formations (§§97, 104), the fundamental formulae for the solution • of plane triangles, we will now pass to the consideration of spherical trigonometry, a subject specially adapted for treatment by qua- ternions ; but to which we cannot afford more than a very few sections. (More on this subject will be found in Chap. X, in con- nexion with the Kinematics of rotation.) The reader is referred to Hamilton's works for the treatment of this subject by quaternion exponentials.' 111. J Let a, /3, y be unit-vectors drawn from the centre to the corners A, JB,C oi a triangle on the unit-sphere. Then it is evident that, with the usual notation, we have (§ 96), Sa^ = — cos c, Sfiy = —cos a, Sya = —cos &, Trap= sine, TF^y = sin«, TFya= sin 3. Also UVafi, UFj3y, UFya are evidently the vectors of the corners of the polar triangle. Hence S. UFa^ UF^y = cos £, &c., TF.UFa^UF^y = BinB, &c. Now (§ 90 (1)) we have SFapFpy = S.aF.^Fpy =:-Sal38fiy + ^^Say. Remembering that we have SFa^F^y = TFa^TF^yS.UFapUF^y, we see that the formula just written is equivalent to sin a sin c cos B — ■— cos a cos c + cos h, or cos h = cos a cos c + sin a sin o cos B. 112.] Again, F.Fa^F^y = -fiSa^y, which gives TF. FapF^y = S.apy = S.aFfiy = S.^Fya = S.yFa^, or sin a sin csinB = sin a sin^„ = sin b sin p^ = sin c sinjO„ ; where ^„ is the arc drawn from A perpendicular to BC, &c. 113.] INTERPRETATIOKS AND TIUNSFOEMATIONS. 57 Hence sin jo„ = sin e sin £, sin a sin c . _ sm Ml = -. — 5 — sm />, smo sin^o = sin a sin S. 113.] Combining the results of the last two sections, we have Va^.V^y = sin a sin c cos 5— ^ sin a sine sin 5 = sina sine (cos^— /3 sin 5). Hence U. Va^ V^y — (cos 5—^3 sin B), 1 and U. Fy^r^a = (cos ^+ i3 sin B). ) These are therefore versors which turn the system negatively or positively about 0£ through the angle £. As another instance, we have sin 5 tan^ = cos 5 _ Tr.Va^r^y ~ S.Va^r^y _ r.ra^rfiy '^ s.ra^r^y Say + SafiSfiy The interpretation of each of these forms gives a different theorem in spherical trigonometry. Again, let us square the equal quantities F. ajSy and cuS^y— jSSay + ySa^, supposing a, jS, y to be any unit- vectors whatever. We have -{KajSyY = S^^y + S^ya + S^afi+2SfiySyaSafi. But the left-hand member may be written as T\al3y-S^.a^y, whence 1-S^.a^y = S^fiy + S^ya + S^afi + 2S^ySyaSa^, ■ or 1 — cos^fl! — cos^S — cos^c + 2 cos a cos i cos c = sin^a sin^jo„ = &c. ^ sin^asin^3sin^C= &c., all of which are well-known formulae. Such results may be multiplied indefinitely by any one who has mastered the elements of quaternions. 58 QUATERNIOlirS. [114. 114.] A curious proposition, due to Hamilton, gives us a qua- ternion expression for the spherical excess in any triangle. The following proof, which is very nearly the same as one of his, though by no means the simplest that can be given, is chosen here because it incidentally gives a good deal of other information. We leave the quaternion proof as an exercise. Let the unit- vectors drawn from the centre of the sphere to A, B, C, respectively, be a, p, y. It is required to express, as an arc and as an angle on the sphere, the quaternion The figure represents an orthographic projection made on a plane perpendicular to y. Hence G is the centre of the circle BEe. Let the great circle through A, B meet BBe in E, e, and let BB be a quadrant. Thus 2?^ represents y (§ 72). Also make BF=AB=pa~\ Then, evidently, ^ ^ ^a-^y, which gives the arcual representation required. Let BF cut Be in G. Make Ga = EG, and join B, a, and a, F. Obviously, as B is the pole of Ee, Ba is a quadrant ; and since EG — Ca, Ga = EG, a quadrant also. Hence a is the pole oi BG, and therefore the quaternion may be represented by the angle BaF. Make C6 = Ga, and draw the arcs P«/3, Pba from P, the pole of AB. Comparing the triangles Eba and ea(3, we see that Ea = e/3. But, since P is the pole of AB, F^a is a right angle : and therefore as i''a is a quadrant, so is F^. Thus AB is the complement of Ba. or ySe, and therefore „o _ lAB. 1 1 5. J INTEEPEETATIONS AND TRANSFOEMATIONS. 59 Join bA and produce it to c so tnat Ac = hA; join e, P, cutting AS in 0. Also join c, £, and £, a. Since Pis the pole of AS, the angles at o are right angles ; and therefore, by the equal triangles 6aA, go A, we have aA = Ao. But a^ = 2AB, whence oB = B^, and therefore the triangles coB and Bafi are equal, and c, ^, a lie on the same great circle. Produce cA and cB to meet in M (on the opposite side of the sphere). H and c are diametrically opposite, and therefore cP, produced, passes through H. Now Pa = Pb = PH, for they differ from quadrants by the equal arcs fl/3, ba, oc. Hence these arcs divide the triangle Eab into three isosceles triangles. But IPHb + IPHa = LaHb = Ibca. Also /.Pab = TT—Zcab — Z-PaH, LPba =. LPab = it- Lcba- LPbH. Adding, iLPab^lis— Leah — Lcba— Lbca = IT — (spherical excess oi abc). But, as LFaj3 and LBae are right angles, we have angle of /3a~V = ^^aJ) = L^ae — LPab = \ (spherical excess, of abc). [Numerous singular geometrical theorems, easily proved ab initio by quaternions, follow from this : e. g. The arc AB, which bisects two sides of a spherical triangle abc, intersects the base at the distance of a quadrant from its middle point. All spherical tri- angles, with a common side, and having their other sides bisected by the same great circle (i. e. having their vertices in a small circle parallel to this great circle) have_equal areas, &e., &c.J 115.] Let 0« = a, Ob = /3', Oc = y', and we have ©^ey(jr=^-^^-^^ ^^V Vy'. = Ca.BA But FQ is the complement of BF. Hence the angle of the quaternion , a A ^ /S'v I / /v Kj') yz') \7) 60 QUATERNIONS. [ll6. is half the spherical excess of the triangle whose angular points are at the extremities of the unit-vectors a', ^', y' . [In seeking a purely quaternion proof of the preceding proposi- tions, the student may commence by shewing that for any three unit- vectors we have a.,„ The angle of the first of these quaternions can be easily assigned ; and the equation shews how to find that of /Sa-^y. But a stUl simpler method of proof is easily derived from the composition of rotations.] 116.] A scalar equation in p, the vector of an undetermined point, is generally the equation of a surface; since we may sub- stitute for p the expression . _ ^^j where x is an unknown scalar, and a any assumed unit-vector. The result is an equation to determine x. Thus one or more points are found on the vector xa whose coordinates satisfy the equation j and the locus is a surface whose degree is determined by that of the equation which gpives the values of x. But a vector equation in p, as we have seen, generally leads to three scalar equations, from which the three rectangular or other components of the sought vector are to be derived. Such a vector equation, then, usually belongs to a definite number oi points in space. But in certain cases these may form a line, and even a surface, the vector equation losing as it were one or two of the three scalar equations to which it is usually equivalent. Thus while the equation ap — & gives at once p _ „-i^^ which is the vector of a definite point (since we have evidently /Sa/3 = 0) ; the closely allied equation y^^ _ a is easily seen to involve g^o _ q and to be satisfied by p — oT'^R+xa whatever be x. Hence the vector of any point whatever in the line drawn parallel to a from the extremity of a~^/3 satisfies the given equation. 117.] Again, Fap .Fp^ = {FafiY is equivalent to but two scalar equations. For it shews that Fap 119.] INTERPRETATIONS AND TRANSFORMATIONS, 61 and F)3p are parallel, i. e. p lies in fhe same plane as a and (3, and can therefore be written (§ 24) p = asa+^A where x and _y are scalars as yet undetermined. We have now Fap = yVafi, which, by the given equation, lead to xy =■ \, or y = -, or finally p = xa+~j3i w which (§ 40) is the equation of a hyperbola whose asymptotes are in the directions of a and ^8. 118.] Again, the equation r.raprap = o, though apparently equivalent to three scalar equations, is really equivalent to one only. In fact we see by § 91 that it may be written -aS.a^p = 0, whence, if a be not zero, we have S.ajSp = 0, and thus (§101) the only condition is that p is coplanar with a, j3. Hence the equation represents the plane in which o and )3 lie. 119.] Some very curious results are obtained when we extend these processes of interpretation to functions of a quaternion, q = w+p instead of functions of a mere vector p. A scalar equation containing such a qtiaternion, along with quaternion constants, gives, as in last section, the equation of a surface, if we assign a definite value to w. Hence for successive values of w, we have successive surfaces belonging to a system; and thus when w is indeterminate the equation represents not a surface, as before, but a volume, in the sense that the vector of any point within that volume satisfies the equation. Thus the equation {Tqf = a^, or w'^—p^ = a^, or ' {Tpf = a^-w^, represents, for any assigned value of w, not greater than a, a sphere whose radius is ^/a^ — w^. Hence the equation is satisfied by the 62 QUATEENIONS. [l20. vector of any point whatever in the volume of a sphere of radius a, whose centre is origin. Again, by the same kind of investigation, where q = w + p, is easily seen to represent the volume of a sphere of radius* a: described about the extremity of ^ as centre. Also S{^)-= —a? is the equation of infinite space less the space contained in a sphere of radius a about the origin. Similar consequences as to the interpretation of vector equations in quaternions may be readily deduced by the reader. 120.] The following transformation is enuntiated without proof by Hamilton {Lectures, p. 587, and Elements, p. 299). »--i(rY)*5-i = U{rq-\-KrKq). To prove it, let r~\r^g^)^g~^ = t, then Tt = 1, and therefore But {r^ff = rti, or r'^q^ = rtqrtq^, or rq^ = tgrt. Hence KqKr - t-'^KrKqr\ or KrKq = tKqKH. Thus we have jji^^^ + ^^^^j = tU{qr±KqKr) t, or, if we put * = U{qr + KqKr), Ks= ± Ut. Hence sKs = {Tsf = 1 = ± stst, which, if we take the positive sign, requires st= ±\, or t= +«-!= ±UKs, which is the required transformation. [It is to be noticed that there are other results which might have been arrived at by using the negative sign above ; some in- volving an arbitrary unit- vector, others involving the imaginary of ordinary algebra.J 121.] As a final example, we take a transformation of Hamil- ton's, of great importance in the theory of surfaces of the second order. 121.] INTERPRETATIONS AND TRANSFORMATIONS. 63 Transform the expression • in which a, 13, y are any three mutually rectangular vectors, into the form mt , \ 2 MW + PkV ^ which involves only two vector-constants, t, k. {T{ip + pK)}^ = {tp+pK){pi + Kp) (§§ 52, 55) = (l^ 4- K2)p2 + (tpKp +p/Cpt) = {l^+K^)p^+2S.lpKp = {l-K)Y+4:SipSKp. Hence (Sapf + iS^pf+iSypy^^^.p^ + i-^^'P^'P - (^2-12)2'' ' ^(^2-12)2 But a-2(5ap)2 + y3-2(<S'j3/))2 + y-2(;S'yp)2 = p2 (§§ 25, 73). Multiply by jB^ and subtract, we get The left side breaks up into two real factors if ^2 be intermediate in value to a^ and y^ : and that the right side may do so the term in p2 must vanish. This condition gives ft— k)2 fl2 = A^ L^ ■ and the identity becomes ^(aV(l-5) + yV(^-l))p^(aV(l-5)^-yV(^-l)> = 4^. Hence we must have lL^^=^(a^il-.^) + yV{^-l)}, where ^ is an undetermined scalar. To determine j9, substitute in the expression for p^, and we find = {P^ + -^)(a^-7')-2(tt^+y^) t4|32. 64 QUATERNIONS. [l22. Thus the transformation succeeds if 1 2(a'= + /) 1 / o? which gives jo+ - = + 2^/ 2_^ ' 1 v^ Hence J^^ = (jj-y) (a^-/) = ± 4^/^^ or (/c2-i2)-i= iTayy. . . Ta^Ty 1 2'a-yy ^^^"^' ^ = 77=^' 'P=W^^' and therefore ro-I-y , , ^2_a2 y2_^2 Thus we have proved the possibility of the transformation, and determined the transforming vectors i, k. 123.] By diflFerentiating the equation we obtain, as will be seen in Chapter IV, the following, where p also may be any vector whatever. This is another very important formula of transformation ; and it will be a good exercise for the student to prove its truth by processes analogous to those in last section. We may merely observe, what indeed is obvious, that by putting p'= p it becomes the formula of last section. And we see that we may write, with the recent values of i and k in terms of a, /3, y, the identity aSap + l3S^p + ySyp = ^ \J_,2yL _ (t — Kfp + 2 {iSkp + kSip) 123.] In various quaternion investigations, especially in such as involve imaginary intersections of curves and surfaces, the old imaginarj' of algebra of course appears. But it is to be particularly 124-] INTERPRETATIOlirS AND TEANSFOEMATIONS. 65 noticed that this expression is analogous to a scalar and not to a veetorj and that like real scalars it is commutative in multiplica- tion with all other factors. Thus it appears, by the same proof as in algebra, that any quaternion expression which contains this imaginary can always be broken up into the sum of two parts, one realj the other multiplied by the first power of v— 1. Such an expression, viz. ? = /+ V^?", where ({ and /' are real quaternions, is called a biquaternion. Some little care is requisite in the management of these expressions, but there is no new diflBeulty. The points to be observed are : first, that any biquaternion can be divided into a real and an imaginary part, the latter being the product of \/— 1 by a real quaternion ; secondj that this \/ — 1 is commutative with all other quantities in multiplication ; tbirdj that if two biquaternions be equal, as we have, as in algebra, /= /, j"= /'j so that an equation between biquaternions involves in general eight equations between scalars. Compare § 80. 124.] We have, obviously, since ^/— i is a scalar, Hence (§103) = {8q'+^^-i.Sf+ ?Y+ ^/irTr/')('S/+ V^^/'- rq'- sf- 1 Yf) = (Sq'+ ,y^I\Sff-{rq'+ ^/^^/')^ = {Tq'f - {Tff + 2 aA^aS. ^Kf. The only remark which need be made on such formulae is this, that the tensor of a hiquaternion may vanish while both of the component quaternions are finite. Thus, if ^/= Tq", and S.q'Kq"= 0, the above formula gives The condition S.^Kq"= may be written Kq"=q'-^a, or q"= -aKq'-^=- ^^r where a is any vector whatever. 6 6 QUATERNIONS. [ 1 2 5 . Henee Tq' = Tq" = TKq" = ^ , and therefore Tq\Uq'- </::::\Ua.U^) = (l - ^/^^Ua)^ is the general form of a biquaternion whose tensor is zero. 125.] More generally we have, q, r, ^, / being any four real and non-evanescent quaternions, {qJr '/^cf) (r+ ^/^T/) = qr-c['/+ ^^Ix^q/ Jf^r). That this product may vanish we must have qr = q'/,- and q/= —q'r. Eliminating / w;e have qq'~^qr = — /?', which gives {l'~^s)^ = ~^> i.e. q = ({a where a is some unit-vector. And the two equations now agree in giving — r = a/, so that we have the biquaternion factors in the form /(a+V^) and — (a-^/^)/; and their product is -/(a+ ^T-i) (a- sT^y, which, of course, vanishes. [A somewhat simpler investigation of the same proposition may be obtained by writing the biquaternions as g^C^-^^+y^) and (?-/-i+^/3i)/, or g'(/'+V^) and (Z'+v'ZIT)/, and shewing that 5"= — /'= a, where Ta = 1.] From this it appears that if the product of two biveciors p + trV — l and p' + ff'v— 1 is zero, we must have ^-ip = _pV-i = Ua, where a may be any vector whatever. But this result is still more easily obtained by means of a direct process. 126.] It may be well to observe here (as we intend to avail our- selves of them in the succeeding Chapters) that certain abbreviated 127.] INTERPRETATIONS AND TRANSFORMATIONS. 67 forms of expression may be used when they are not liable to confuse, or lead to error. Thus we may write T^q for {Tqf, just as we write ^os^fl for (eos Of, although the true meanings of these expressions are T{Ta) and cos (eos 0): The former is justifiable, as T{Ta) = Ta, and therefore T^d is not required to signify the second tensor (or tensor of the tensor) of a. But the trigonometrical usage is quite indefensible. Similarly we may write S^q for {Sqf, &c., but it may be advisable not to use Sq^ as the equivalent of either of those just written ; inasmuch as it might be confounded with the (generally) different quantity S.q^ or S{q^), although this is rarely written without the point or the brackets. 137.] The beginner may expect to be a little puzzled with the aspect of this notation at first ; but, as he learns more of the sub- ject, he will soon see clearly the distinction between such an ex- pression as S.FapriSy, where we may omit at pleasure either the point or the first F with- out altering the value, and the very different one Sa^.rpy, which admits of no such changes, without altering its value. All these simplifications of notation are, in fact, merely examples of the transformations of quaternion expressions to which part of this Chapter has been devoted. Thus, to take 3. very simple ex- ample, we easily see that S.Va^r^y = SFapr^y = S.a^FjSy = SaF.^Ffiy = -SaF.{Ffiy)p = SaF.{Fy^)P = S.aF{yP)^ = S.F{yP)pa = SFy^F^a = S.y^F^a = &c., &c. The above group does not nearly exhaust the list of even the simpler ways of expressing the given quantity. We recommend it to the careful study of the reader. He will find it advisable, at first, to use stops and brackets pretty freely ; but will gradually learn to dispense with those which are not absolutely necessary to prevent ambiguity. F 2 68' QUATERNIONS. EXAMPLES TO CHAPTER III. 1. Investigate, by quaternions, the requisite formulse for changing from any one set of coordinate axes to another; and derive from your general result, and also from special investigations, the usual expressions for the following cases : — (a.) Rectangular axes turned abbut « through any angle. (b.) Rectangular axes turned into any new position by rota- tion about a line equally inclined to the three. (c. ) Rectangular turned to oblique, one of the new axes lying in each of the former coordinate planes. 2. If Tp = Ta = T^ = 1, and S.a^p = 0, shew by direct transfor- mations that ^_ jj^p _ „^ j;r(p _^) ^ + ^in-SalB). Interpret this theorem geometrically. 3. If Sa^ = 0, Ta=T^=l, shew that (1 +0™)^ = 2 cos^a^;8 = 2Sa^.a^^. 4. Put in its simplest form the equation pS. Fa^ r^y Fya = aV. Fya Fafi + 6F. Fafi FjSy + c F. Ffiy Fya ; and shew that a = S.fiyp, &c. 5. Prove the following theorems, and exhibit them as properties of determinants : — {a.) S.[a + ^){fi + y){y^a) = 2S.apy,- {h.) S.Fa^F^yFya = -(S.a^y)'^, (c.) S.F(a + l3)i^ + y)F{l3 + y){y + a)F{y + a){a+p) = -4(5.a/3y)^ (d.) S. F( Fafi Ffiy) F( Fj3y Fya) F{ Fya Fa^) = - {S.a^y)\ {e.) S.5€C = — \6{8.afiy)*, where b = F{Fia+l3){^ + y)F(l3 + y)(y + a)), t = F{FiP+y){y + a)F(y+a)(a + p)), {:=F{F(y + a)(^a + l3)F{a + l3)(l3 + y)). 6. Prove the common formula for the product of two determi- nants of the third order in the form S.a^yS.a^^iyi^ — Saa^ <S/3aj Sya^ Safi, mi3i Syfi, Sayi Si3yi Syy^ 7. If, in § 102, a, j8, y be three mutually perpendicular vectors, can anything be predicted as to Oi, jSj, yj ? If a, j3, y be rectangular unil vectors, what of Oj, p^, y^? EXAMPLES TO CHAPTER III. 69 8. If aj /3, y, a', 13', y be two sets of rectangular unit-vectors^ shew that Saa'= Syfi'SjSy'-S^fi'Syy', Sec, &c. 9. The lines bisecting pairs of opposite sides of a quadrilateral are perpendicular to each other when the diagonals of the quadri- lateral are equal. 10. Shew that (6.) S.q^=S^q-3SqT^rq, (e.) a^p^y^+S^al3y = r\afiy, (d.) S{r.a^yF.Pyar.yal3) = 4: Sa^S^ySyaS.a^y, (e.) r.q^= (3 S^q-T'' Vq) Yq, (/.) qVYq-^ = -Sq.Urq + TFq; and interpret each as a formula in plane or spherical trigonometry. 11. If g- be an undetermined quaternion, what loci are repre- sented by (a.) {qa-^r = -a^ {b.) {qa-^Y=a\ {e.) S.{q-aY=a\ where a is any given scalar and a any given vector ? 12. If ^ be any quaternion, shew that the equation is satisfied, not alone by Q,= ±q but also, by Q = + ^/~:^{Sq.JJVq-TYq). (Hamilton, Lectures, p. 673.) 13. Wherein consists the difference between the two equations T^^=l, and (^^=-1? a ^a' What is the full interpretation of each, a being a given, and p an undetermined, vector ? 14. Find the full consequences of each of the following groups of equations, both as regards the unknown vector p and the given vectors a, /3,y:— „ „ on Sap = 0, Sap = 0, («•) of'' " I' (*•) ^•''^P = '^' ^'-^ ^•"^'' = ^' S.pyp = 0; g^^ ^Q. S.a^yp = 0. 15. From §§ 74, 109, shew that, if e be any unit-vector, and m any scalar, c" = cos — + e sm — • 70 QUATERNIONS. Hence shew that if a, j3, y be radii drawn to the corners of a tri- angle on the unit-sphere, whose spherical excess is m right angles, /3 + y'a+/3'y + a Also that, if A, B, C be the angles of the triangle, we have i£ iB iA y" ^"a" = — 1. 16. Shew that for any three vectors o, j3, y we have {Ua^)^ + {UpY)'^+{Uayy + {U.a^yy + iUay.SUa^SUpy = -2. (Hamilton, Elements, p. 388.) 17. If «i, Og, ag, OS be any four scalars, and p-^, p^, pg any three vectors, shew that {8.p^^P^f+{^.a^rp^P^y+ic^{-^rp,p,f-x\^.a^{p,-p,)y + 2n(aj2 + Spyp^ + ttjO,^ = 2n(a!2 -f p^) + 2n«'' + 2{(a!2 +%'' +Pi^) ((^p^pg)'' + 2 «A(aj2 + Sp^pg) -a!^(p2-ps)^)} ; where Yla^ = a^a^a^. Verify this formula by a simple process in the particular case «j = 02 = 03 = a; = 0. {Ibid) CHAPTER IV. DIFFERENTIATION OF QUATERNIONS. 128.] In Chapter I we have already considered as a special case the differentiation of a vector function of a scalar independent variable: and it is easy to see at once that a similar process is applicable to a quaternion function of a scalar independent variable. The differential, or differential coefficient, thus found, is in general another function of the same scalar variable ; and can therefore be differentiated anew by a second, third, &c. application of the same process. And precisely similar remarks apply to partial differentia- tion of a quaternion function of any number of scalar independent variables. In fact, this process is identical with ordinary differ- entiation. 129.] But when we come to differentiate a function of a vector, or of a quaternion, some caution is requisite ; there is, in general, nothing which can be called a differential coefficient ; and in fact we require (as already hinted in § 33) to employ a definition of a differential, somewhat different from the ordinary one but, coinciding with it when applied to functions of mere scalar variables. 130.] If r=F{q) be a function of a quaternion q, d^ = dFq = ^^n {F{q + '^±)-F{q)}, where » is a scalar which is ultimately to be made infinite, is defined to be the differential of r or Fq. Here dq may be any quaternion whatever, and the right-hand member may be written /., g s where / is a new function, depending on the form of F; homo- geneous and of the fi,rst degree in dq ; but not, in general, capable of being put in the form f ^^) j^_ 7 2 QUATERNIONS. [ 1 3 1 . 131.] To make more clear these last remarks, we may observe that the function y/„ g^ thus derived as the differential of V{q), is distributive with respect to dq. That is y (^^ ^ + ,) = y (^, ^) + y (^, ,)^ r and « being any quaternions. For /(?, r + *) = ^^ « (i? (^ + ^) - i^-C?)) And, as a particular case, it is obvious that if a; be any scalar /fe <»r) = isfiq, r). 132.] And if we define in the same way dF{q,r,s ) as being the value of ■C.«|'(s+*' '+*••+*• )-^(^.'.'. )}■ where q,r,Sy... dq, dr, ds, are any quaternions whatever ; we shall obviously arrive at a result which may be written f{q, r, s, ...dq, dr, ds, ), where ./ is homogeneous and linear in the system of quaternions dq, dr,ds, and distributive vrith respect to each of them. Thus, in differentiating any power, product, &c. of one or more quater- nions, each factor is to be differentiated as if it alone were variable ; and the terms corresponding to these are to be added for the com- plete differential. This differs from the ordinary process of scalar differentiation solely in the fact that, on account of the non-com- mutative property of quaternion multiplication, each factor must in general be differentiated in situ. Thus d{gr) = dq.r + qdr, but not generally = rdq + qdr. 133.] As Examples we take chiefly those which lead to results which will be of constant use to us in succeeding Chapters. Some of the work will be given at full length as an exercise in quaternion transformations. (1) {Tpf=-p^. The differential of the left-hand side is simply, since Tp is a scalar, 2TpdTp. 1 3 3- J DltFEEENTIATION. 73 That Of p^ is ^^n((^p + ±f -p^) = 2Spdp. Hence Tp dTp = -Spdp, or dTp=-S.Updp = sf'' dTp ^dp or -=i- = ;iS — Up' (2) Again, p = TpUp dp = dTp.Up + TpdUp, , dp dTp dUp whence JL-iLj^i^ p Tp Up = .i + f by(.). Hence dUp _ -p-dp W~ J' This may be transformed into F-^ or -^-^ » &e. p2 Tp^ (3) iTqy = qKq 2TqdTq = i(^X^) = ^^n^(q + ^J)K(q + ^) -qKq], = l.-(&±Mi^^,^Kdq), = qKdq + dqKq, = qKdq + K{qKdq) (§55), = iS.qKdq = iS.Kqdq. Hence dTq = S.UKqdq = S.Uq-'^dq since :Z^ = :?'% and 27X^ = ?7^-i. If 3' = p, a vector, Kq = Kp = —p, and the formula becomes dTp = —S. Updp, as in (1). Again, dTq dq Tq-q But dq=TqdUq+UqdTq, which gives dq dTq dUq q- Tq^ Uq' whence, as dq_dTq q Tq we have dq _ dUq f —^ i Uq 74 QUATERNIONS. [134. 2 (4) aif)=<..^(ii+^y-f) = qdq + dq.q = 2S.qdq + 2Sq.Fdq + 2Sdq.Vq. If g' be a vector, as p, Sq and Sdq vanisli, and we have d{p^) = 28pdp, as in (1). (5) Let q = r*. This gives dr^ = dq. But ^ = d{q^) = qdq + dq.q. This, multiplied iy ^ and m^o Kq, gives and drKq = dq.Tq^+qdq.Kq. Adding, we have qdr + dr.Kq = {q^ + Tq^ + 2<%.j) <«j ; whence dq, i. e. <^^, is at once found in terms of dr. This process is given by Hamilton, Lectures, p. 628. (6) qq-^ = 1, qdq~^ + dq.q~^ = ; . • . dq-"^ = — q-^ dq.q-^. If gf is a vector, = p suppose, dq~^ = —p~^dp.p~^ p^ p p (7) q = Sq+Fq, dq= dSq + dFq. But dq = ^^j- + Fi:?^. Comparing, we have dSq = Sdq, dVq = Vdq. Since Xq = Sq— Vq, we find by a similar process <?X2 = Kdq. 134.] Successive diflFerentiation of course presents no new dif- ficulty. Thus, we have seen that d{q^) = dq.q + qdq. 1 35-] DIFFEEENTIATION. 75 DiflFerentiating again, we have and so on for higher orders. If §' be a vector, as p, we have, §133(1), d{p^) = 2Spdp. Hence d^(p^) = 2{dpf + 2Spd^p, and so on. Similarly d^Up= -dA-Fpdp) • But d <Tp^ 1 _ 2dTp 2Spdp Tp"^ ~ Tp^ ~ Tp*^ and d. Vpdp = V. pd^p. Hence -^^J^p =- ^(rpi,)^+ Wp ^ 2J^^^^ = - ^ ((^P^P)' +P' Fp^V- 2 Fp^p^p^p) * 135.] If the first differential of q be considered as a constant quaternion, we have, of course, d^q = 0, d^q = 0, &e., and the preceding formulae become considerably simplified. Hamilton has shewn that in this case Taylor's Theorem admits of an easy extension to quaternions. That is, we may write f{q + xdq) =/{q) + xd/{q) + ~ d^iq) + if d'^q = ; subject, of course, to particular exceptions and limita- tions as in the ordinary applications to functions of scalar variables. Thus, let y($') = q^) and we have 4f(q) = q^dq + qdq.q + dq.q^, d^/iq) = 2dq.qdq + 2q{dq)^ + 2idq)''q, d^f{q) = G{dq)\ and it is easy to verify by multiplication that we have rigorously (g- + xdqf= f + x{q^dq + qdq.q + dq.q^) + x" {dq.qdq 4 q {dqf + {dqfq) + a;^(dqf ; which is the value given by the application of the above form of Taylor's Theorem. As we shall not have occasion to employ this theorem, and as the demonstrations which have been found are all too laborious for an elementary treatise, we refer the reader to Hamilton's works, where he will find several of them. * This may be farther simplified ; but it may be well to caution the student that we cannot, for such a purpose, write the above expression as -^J.pidpYpdp + d'p.p-'- 2dpSpdp}. 76 QUATERNIONS. [136. 1 36.] To differentiate a function of a function of a quaternion we proceed as with scalar variables, attending to the peculiarities already pointed out. 137.] A case of considerable importance in geometrical appli- cations of quaternions is the differentiation of a scalar function of p, the vector of any point in space. Let F{p) = C, where i^ is a scalar function and C an arbitrary constant, be the equation of a series of surfaces. Its differential, f{p, dp) = 0, is, of eourscj a scalar function : and, being homogeneous and linear in dp, § 130, may be thus written, Svdp = 0, where i; is a vector, in general a function of p. This vector, v, is easily seen to have the direction of the normal to the given surface at the extremity of p ; being, in fact, per- pendicular to every tangent line dp, §§ 36, 98. Its length, when F is a surface of the second degree, is as the reciprocal of the distance of the tangent-plane from the origin. And we will shew, later, that if p = ix+jy+&z, / . d . d , d \ „ EXAMPLES TO CHAPTER IV. 1 . Shew that (a.) d.SUq = s.Usr^=-s^Truq, (b.) d.rUq=r.Uq-^F^dq.q-^), (c.) d.TrUq = S^=:S^^SUq, {d.) d.a" = ^ a^+'^dm, (e.) d\Tq={^.dqq-^-S.{dqq-^f}Tq = -~Tqr^^' 2. If Fp='2.Sap8l3p+iffp^ give dFp t= Svdp, shew that v = S T. ap^ + (^ + 2 Sa^) p. CHAPTER V. THE SOLUTION OP EQUATIONS OF THE PIEST DEGEEE. 138.] We have seen that the differentiation of any function whatever of a quaternion^ q, leads to an equation of the form where/" is linear and homogeneous in dq^. To complete the process of differentiation, we must have the means of solving this equation so as to be able to exhibit directly the value of dq. This general equation is not of so much practical importance as the particular case in which dq is a . vector ; and, besides, as we proceed to shew, the solution of the general question may easily be made to depend upon that of the particular case j so that we shall commence with the latter. The most general expression for the function _/ is easily seen to be dr =/(§■, dq) = 2 V.adqh + S.cdq, where a, I, and c may be any quaternion functions of q whatever. Every possible term of a linear and homogeneous function is re- ducible to this form, as the reader may easily see by writing down all the forms he can devise. Taking the scalars of both sides, we have Sd^- = S.cdq = SdqSa + S.rdqFc. But we have also, by taking the vector parts, Fd?- = 2 r. adqb = Sdq.^ rab + -2,r.a{ Vdq) b. Eliminating Sdq between the equations for Sdr and Vdr it *is obvious that a linear and vector expression in Vdq will remain. Such an expression, so far as it contains Vdq, may always be reduced to the form of a sum of teims of the type aS.^Vdq, by the help of formula like those in §§ 90, 91. Solving this, we have Tdq, and Sdq is then found from the preceding equation. 78 QUATERNIONS. [139. 139.] The problem may now be stated thus. Find the value of p from the equation o5/3p+ai-S)3ip+ ... = 2.aSfip = y, where a, 13, a^, ^i, ...y are given vectors. [It will be shewn later that the most general form requires but three terms, i. e. six vector constants a, y3, a^, ^j, Og, /Sg in all.] If we write, with Hamilton, (j>p = 2.a<S)3p, the given equation may be written <pp = y. or p = (j>-^y, and the object of our investigation is to find the value of the in- verse function (jr'^, 140.] We have seen that any vector whatever may be expressed in terms of any three non-coplanar vectors. Hence, we should ex- pect a priori that a vector such as <p(p4>p, or <j)^p, for instance, should be capable of expression in terms of p, <j)p, and (p^p. [This is, of course, on the supposition that p, (j)p, and (fi^p are not generally co- planar. But it may easily be seen to extend to this case also. For if these vectors be generally coplanar, so are <j)p, (p^p, and <j)^p, since they may be written <r, ifxr, and (/)V. And thus, of course, ^^p can be expressed as above. If in a particular case, we should have, for some definite vector p, <pp=gp where ^ is a scalar, we shall obviously have <^^p =g^p and ^^p =g^p, so that the equation will still subsist. And a similar explanation holds for the particular case when, for some definite value of p, the three vectors p, Kpp, <^^p are coplanar. For then we have an equation of the form ^^p = Ap-i- Bijip, which gives (l>^p = A(l>p + £(l)^p = ABp-\-{A + B^)<i>p. So that (p^p is in the same plane.] If, then, we write -(t,^p = xp+y4>p + e(l)^p, (1) it is evident that x, y, z are quantities independent of the vector p, and we can determine them at once by processes such as those iu §§91,92. If any three vectors, as «', /, h, be substituted for p, they will in general enable us to assign the values of the three coeflScients on 142.] SOLUTION OF EQUATIONS, 79 the right side of the equation, andme solution is complete. For by putting (t>~^p for p and transposing, the equation becomes that is, the unknown inverse function is expressed in terms of direct operations. If x vanish, while y remains finite, we substitute ^~V for p, and have -y (^-^ = «p + cj,p, and if x and _y both -vanish — Z(j>~^p = p. 141.] To illustrate this process by a simple example we shall take the very important case in which <f) belongs to a central surface of the second order ; suppose an ellipsoid ; in which case it will be shewn (in Chap. VIII.) that we may write ^p = —a^iSip — h^jSjp—c^JcSkp, Here we have ipi = cp'i, <^H = aH, <f)H = a^i, 4,j = by, <t>y = b*j, <t>y = by, (pk = cH, ^H = c*/i, ^^k = o^h. Hence, putting separately i,j, Tc for p in the equation (1) of last section, we have —a^ = x^ya^-\-m^, —b^= le+yb^+zb*, — C® = X-\-i/C^ +ZC*. Hence a^, b^, c^ are the roots of the cubic ^* + «P +.?'£+«= 0, which involves the conditions z=-{a^ + l^ + c^), y = cfib"^ + b'^c^ + c^a^, x = — a^b^c^. Thus, with the above value of ^, we have (/>3p = aWc^p - {aW + h^c^ + c V) # + {a^ + b^-\- c^) <p^p. 142.] Putting ^"^(T in place of p (which is any vector whatever) and changing the order of the terms, we have the desired inversion of the function ^ in the form aWc^-'^a- = {aW + bH^ + (^a^) a—{a'^ + b^ + c^) (fxr + ^V, where the inverse function is expressed in terms of the direct func- tion. For this particular case the solution we have given is com- plete, and satisfactory; and it has the advantage of preparing the reader to expect a similar form of solution in more complex cases. 80 QUATERNIONS. [143. 143.] It may also be useful as a preparation for what follows, if we put the equation of § 141 in the form = *(p') = 4,^p-{a^ + 6^ + c^)(l>''p + {aH'^ +¥c^ +c^a^)^p-a%^c^ p = {(«^-«') (</>-*') (<^-«')}p (2) This last transformation is permitted because </> is commutative with scalars like a*, i. e. <p{a^p) = a^^p. Here we remark that (by § 140) the equation r.p0p = 0, or ^p = gp, where g is some undetermined scalar, is satisfied, not merely by every vector of null-length, but by the definite system of three rect- angular vectors Ai, Bj, Ck whatever be their tensors, the corre- sponding particular values of g being a^, h^, c^. 144.J We now give Hamilton's admirable investigation. The most general form of a linear and vector function of a vector may of course be written as </)p = 'S.V.qpr, where q and r are any constant quaternions, either or both of which may degrade to a scalar or a vector. Hence, operating by S.a- where o- is any vector whatever, S(r(l>p = 2ScTF.qpr = '28pF.raq = 8p4)'(T, (3) if we agree to write ^'o- = IiF.raq, and remember the proposition of § 88. The functions <^ and <j/ are thus conjugate to one another, and on this property the whole in- vestigation depends. 145.] Let A, p. be any two vectors, such that ^p ^ Vkp,. Operating by SX and S.p. we have 8k<^p = 0, Sp.(t>p = 0. But, introducing the conjugate function <^', these become Sp(f>'K = 0, Sp^'p. = 0, and give p in the form mp = Fcjt'kcli'p,, where mis a scalar which, as we shall presently see, is independent of A, jM, and p. But our original assumption gives p = <(>-W\ix; hence we have m^~Wkp. = F^'k(p' p., (4) and the problem of inverting <^ is solved. 147.] SOLUTION OF EQUATIONS. 81 146,] It remains to find the value of the constant m, and to express the vector Vd/kcb'u as a function of FX/n. Operate on (4) by /S.^'r, where v is any vector not coplanar with X and /n, and we get mS.(j/v(l>-^F\n = mS.v<i><irWKix (by (3) of § 144) = mS.Kixv = S.^'X^'ji^'v, or m = S.Xfxv (5) J3 q r S.4>'K4>'iJ.(^'v, Pi ix ^1 H 2'2 ^i p q r S.KjxVf Pi Si '•i [That this quantity is independent of the particular vectors \, ju, v is evident froija the fact that if k'=p)\. + qiJL + ri>, i/ = pjk + q-i^ix + r.^v, and d'= j?2^+S'2M+»"2»' be any other three vectors (which is possible since X, [x, v are not coplanar), we have <i)'k'= p<i)KJrq^' !!.-{- r<l>'v, &C., &C.-, from which we deduce and so that the numerator and denominator of the fraction which ex- presses m are altered in the same ratio. Each of these quantities is in fact an Invariant, and the numerical multiplier is the same for both when we pass from any one set of three vectors to another. A still simpler proof is obtained at once by writing A +j3/x for \ in (5), and noticing that neither numerator nor denominator is altered.] 147.] Let us now change ^ to <i>-\-g, where g is any scalar. It is evident that ^' becomes <i>'+g, and our equation (4) becomes mg{4>^-g)-WkiJ,= r{4,''+g)k{<t>'+g)ixi = r<t>'k^'ix+gF((l/k,j. + k<t>',x)+g'rk^, = {'m(t)~^ +gx+g^)V^kix suppose. In the above equation _ S.{cl>',+g)k{ct/+g)t,{^'+g)v '^'- sJili, = m+m^g+m^g^+g^ 82 QUATERNIONS. [148. is what m becomes when ^ is changed into ^-Vg; % and m^ being two new scalar constants whose values are "^ Sl^v ' _ S. {kij.(f>'v + 4>' kfiv + X.(l>'iJLv) If, in these expressions, we put k+pjx for \, we find that the terms in jp vanish identically ; so that they also are invariants. Substi- tuting for Mg, and equating the coefficients of the various powers of ^ after operating on both sides by ^-f-^, we have two identities and the following two equations, % = '^ + X. [The first determines x, and shews that we were justified in treat- ing F{((/\ij,-\-\<f>'^i) as a linear and vector function of F.Xi/,. The result might have been also obtained thus, SAx^Xfi. = S.\<f/\ix=—S.\ix(t/\=-8.\(l)rhiJ., S.fjLx^^fJ^ = S.jjlKcj/ij, = —S.iiipVKjj,, S.vxVXix. = S.{v^'Xii. + vk4>'i).) = m2SKij,v—S.\iJi.^'v = S.v {m^Vkfi—^fKii) ; and all three (the utmost generality) are satisfied by X = %- *-J 148.] Eliminating ^ from these equations we find or m<l)~^ = OTj — ^j (^ 4- (/)^, which contains the complete solution of linear and vector equations. 149.] More to satisfy the student of the validity of the above investigation, about whose logic he may at first feel some diffi- culties, than to obtain easy solutions, we take a few very simple examples to begin with : we treat them with all desirable prolixity, and we append for comparison easy solutions obtained by methods specially adapted to each case. 150.] Example I. Let <l>p = V.apfi = y. Then <^'p = V.^pa = <^p. Hence m = -=r^ — S ( V. aX^ V. au/3 V. av^). 8.\iJ,v ^ ' 1 5 3-] SOLUTION OP EQUATIONS. 83 Now X, n, V are any three non-eoplanar vectors; and we liiay therefore put for them a, ^,y if the latter be non-coplanar. With this proviso % = -s-^'Sf(a2/3.a/3'2.y + a.a/32.r.oy^ + a2/3.j3.r,ay/3) ,2o2 = — O' 1 S.a^y = —Sap. S (Ti^^.yS.y + a.a/32.y + o;8 V.ayfi) Hence which is one form of solution. By expanding the vectors of products we may easily reduce it to the form a^^^Safi.p = - a^/S^ y + a^^Say + Ba^Sfiy, a-^Say + B-^S3y—y or p = — ' ■ — -^ — - ■ 151.] To verify this solution, we have ^•"''^ "= ^O-^ay + a-^/Sy-r.ay/S) = y, which is the given equation. 153.] An easier mode of arriving at the same solution, in this simple case, is as follows : — Operating by S.a and S.p on the given equation r.opjS = y, we obtain a^SjSp = Say, P^Sap = S^y ; and therefore aSfip = a~'^Say, pSap = /8-i/S'/3y. But the given equation may be written aS^p—pSa^ + ISSap = y. Substituting and transposing we get pSafi = a'^Say + p-^S^y—y, which agrees with the result of § 150. 153.] If a, fi, y be coplanar, the above mode of solution is appli- cable, but the result may be deduced much more simply. For (§101) S.aPy = 0, and the equation then gives S.a^p = 0, so that p is also coplanar with a, /3, y. 6 3, 84 QUATEENIONS. [154. and at once „ _ „-i,,o-i Hence the equation may be written app = y, P = a"V^' and this, being a vector, may be written This formula is equivalent to that just given, but not equal to it term by term. [The student will find it a good exercise to prove directly that, if o, /3, y are coplanar, we have ^(a-i/Sfay + ^-i*S/3y-y) = a-'^S^'^y^^-^Sar'^y-ySar'^^-'^r^ The conclusion that o a n b.app = 0, in this case, is not necessarily true if 5a/3 = 0. But then the original equation becomes aS^p + pSap =: y, which is consistent with S.aPy = 0. This equation gives ^("'^-^«^) = «U/y ^A + ^ Say S ' by comparison of which with the given equation we find Sap and S^p. The value of p remains therefore with one indeterminate scalar. 154. J Example II. Let <^p =: V.a^p = y. Suppose a, ;8, y not to be coplanar, and employ them as A, ft, v to calculate the coefficients in the equation for (j)"^. We have S.(T(j)p = S.cra^p = S.pKcra^ = S-pcj/a: Hence <^'p = ^-palS = V.I3ap. We have now = a^fi^Safi, m, = -=-— -(5(0.^0/3. r, /Say + ;3a2./3.r.;3ay + ;3a2./3a/3.y) o.apy = 2{Safif + a^^^, «*2 = "cV S(a.^.r.fiay+a.^a^.y + ^aK^.y) o.apy = 38al3. 156.] SOLUTION OF EQUATIONS. 85 Hence = (2 (<So^)2 + a^j3^) y- 3/Sa/3 V.a^y + V.a^ V.a^y, which, by expanding the vectors of products, takes easily the simpler form a^p2Sa^p ^ a^^2y_^^2s^^ ^ 2^Sa^Say-^a?S^y. 155.] To verify this, operate by F.a/3 on both sides, and we have a^^^Sapr.aPp = a'^^W.afiy- r.a^afi'^Say+2ap^Safi8ay-ao?^'^S^y = a?^^ {a8^y-pSay + ySaP)-{2aSap-^a^)P^Say + 2 afi^Sa^Say—aa^^^S^y = a^p^Sa^.y, or V. afip = y. 156.] To solve the same equation without employing the general method, we may proceed as follows : — y = r. a^p = pSa^ + V. r{a^) p, Operating by S. Fa^ we have S.a^y — S.a^pSa^. Divide this by Sa^, and add it to the given equation. We thus obtain o o y + ^^ =pSal3+ r. Viafi) p + S. r{afi) p, = {Sal3+ral3)p, = a/3p. Hence p = /3-1 a-i (y + -^) , a form of solution somewhat simpler than that before obtained. To shew that they agree, however, let us multiply by a^^^Safi, and we get a^/i^Sa^.p = ^aySa^ + fiaS.a^y. In this form we see at once that the right-hand side is a vector, since its scalar is evidently zero (§ 89). Hence we may write a?^^Sa^.p = r.^aySa^-Va^S.a^y. But by (3) of §91, —yS.ap ra/3 + a/S./S ( Fa^) y + /3/S. F{aP) ay + Fa^S.a^y = 0. Add this to the right-hand side, and we have a^P^Sa^.p = y {{Sa^)^-S.al3Fap)-a {Sa0S^y-^S.^ (Fafi) y) + ;8 {Sal3Say + S.F (afi) ay) . But {Safif-S-a^Fa^ = {Sa^f-{Fa^f = a^^\ Sa^8^y-S.fi{Fa^) y = Sa^Sfiy-SftaS^y + ^^Say = ^^Say SapSay + S.F{aP)ay = SafiSay + SafiSay-a^S^y = 2Sa^Say-a'^S^y; and the substitution of these values renders our equation identical with that of § 154. 86 QUATERNIONS. [157. [If n, /3, y be coplanar, the simplified forms of the expression for p lead to the equation Sap.p-^a-^y = y-a-^Say + 2pSa-^fi-^Say-p-'8l3Y, which, as before, we leave as an exercise to the student.J 157.] Example III. The solution of the equation Tep = y leads to the vanishing of some of the quantities m. Before, how- ever, treating it by the general method, we shall deduce its solution from that of V.a^p = y already given. Our reason for so doing is that we thus have an opportunity of shewing the nature of some of the cases in which one or more of m, m^, m^ vanish; and also of introducing an example of the use of vanishing fractions in quaternions. Far simpler solu- tions will be given in the following sections. The solution of the last- written equation is, § 154, a^^^Sa^.p = a^^^y-a^^Say—^a.'^S^y+2^Sa^8o.y. If we now put o^ = e + e where e is a scalar, the solution of the first-written equation will evidently be derived from that of the second by making e gradually tend to zero. We have, for this purpose, the following necessary transforma- tions : - a2^2 _ „^ x.a^ = (e + e) (e - e) = e^ - e^, a^^Say + ^a?8^y = a^.^Say + jSa.aS^y, = {e + e)fiSay + {e—e)aSPy, = e {^Say + aS^y) + eV.yVa^, = e l^Say + aS^y) + e Tye. Hence the solution becomes (e2_e2)ep = {e^-e.^)y-e{^Say + aS^y)-iryi + 2e^Say, - le^—(^)y + eF.yra^—eryf, = ^e^^i'')y + ery€ + yf'-fSyf, = e^y + eVye — tSye, Dividing by e, and then putting e = 0, we have -eV = rye-<„(^). Now, by the form of the given equation, we see that Sye = 0. Hence the limit is indeterminate, and we may put for it cc, where as is any scalar. Our solution is, therefore, or, as it may be written, since Sye = 0, p = e-i(y + a;). l6o.J SOLUTION OF EQUATIONS. 87 The verification is obvious — for we have ep = y + x. 158.] This suggests a very simple mode of solution. For we see that the given equation leaves Sip indeterminate. Assume, therefore, Sep = x and add to the given equation. We obtain ep =x + y, or p=e-i(y + a,), if, and only if, p satisfies the equation Vep = y. 159.] To apply the general method, we may take e, y and ey (which is a vector) for A, p,, v. We find <l)'p = Vpe. Hence Hence m = 0, 1- ^l=-^,S.{..,y.,^y)=-,^ m„ = 0. or That is. P = e~^y + xe, as before. Our warrant for putting xe, as the equivalent of 0"^ is this : — The equation ^2^ _ q may be written r.eFfcr = = <re^ - tSta. Hence, unless o- = 0, we have o- 1| e = xe. 160.] Example IV. As a final example let us take the most general form of t^, which, as will be soon proved, may be expressed as follows : — <^p = ajS/3p + a-^S^-^p + a^S^^p = y. Here <l>p = ^Sap + ^-^80^ p + ^.^Sa^ p, and, consequently, taking a, Oj , 02, which are in this case non- coplanar vectors, for A, p., v, we have -S.(^Saa + ^^Sa^a + P^Sa^oi) {l3Saa-^ + P-^Saj^aj^+ ...) {pSaa2+ ) 'S. aojOg S.1313,^2 Saa Sa-jO, iSogO Saor^ /SojOj (SogOj Saag /iSa^ag Sa^a^ S.aa^a^ = ^f^{ASaa + A^Sa^a + A,Sa^a), 'S.o 88 QUATERNIONS. [l6o. where A = Sa^aiSa^ai— Sai.aiSaia2 = —S. VojO^ VojO^ A^ = Sa^aiSaa2—Saa,ySa^a3 = —S. Fctg a FioiOg A2 = SacijSa^a2 — Saia^Saa^ r= — S. Vaa^ FajOg. Hence the value of the determinant is — {SaaS. FojO^ Va^a^ + SoyoS. Fa^a Va^a^ + Sa^aS. Faa^ FoiOg) = -SMiFaia^S.aaja^) {by § 92 (3)} = -{S.aa^a^f. The interpretation of this result in spherical trigonometry is very interesting, (^ee Ex. (6) p. 68.) By it we see that Similarly, m, = g 8.[a (0Saai + ^,80.0, + /SgSoaai) {^Saa^ + ^i^ojOa + /SaiSaga,) + &c.] a.aOiO^ = 5 {S.a^Pi (800180102— 8010^8002) + ) = o (5.a/3i3i5.a r.Oi ragOi + ) = - ^ IS.a ( FpPy8. Faoi Foy02 + F^^8. F02O rajOg + Ffi^^2^. Fo^a^ Fa^a^ tf.aoia^ +S.ai{Fl3l3^8.FaoyFo20+ ) + S.a2{Fl3Pi8.FaoyFooi+ )] ; or, taking the terms by columns instead of by rows, = — p 18. F^Pi {a8. Faoy Fa^a^ + 0^8. Faa^ Fa^ + 028. Faa-^ Faa-^ 8.00^02 1 = --^^^lS.Ffi^y{FoayS.oay02) + ], _ = -S{FaoiFpfii+ Fai02F^,p2+ T^V^^^i^)- Again, «2„ = -r S\oaA^Saa2 + ^iSoia2+ ...) + a2o{^8oai-\- ...) + a-^02{pSaa+...)'\, or, grouping as before, = — 8\^{ Foai8ao2 + Fa.^o8aay + Foy028aa) 4- • ■ ■] , 0.00^02 = -^^^S[fi {08.00^02)+ j (§92(4)), = 8(0^+0^^1+02^2)- And the solution is, therefore, (f>-^y8.aOya28.p^l^2 = pi^.aaiaa&^^i^a = y25. Fooy T/S/Sj + ^ySSo^S - <J) V- 163-] SOLUTION OF EQUATIONS. 89 \ [It will be excellent practice for the student to work out in detail the blank portions of the above investigation, and also to prove directly that the value of p we have just found satisfies the given equation.] 161. J But it is not necessary to go through such a long process to get the solution — though it wUl be advantageous to the student to read it carefully — ^for if we operate on the equation by S-OjCt^, S.a^a, and Smo^ we get S.aiOf^aSlSp = S.aiO^y, S.a^aoiSfiip =: S.a^ay, S.aayO^S^^p ^ S.aajy. From these, by § 92 (4), we have at once pS.aojO^S.pPiP^ = Fpi3^8.aajy+ r^^^S.a^a^y + V^^S.a^ay. The student will find it a useful exercise to prove that this is equi- valent to the solution in § 160. To verify the present solution we have = a'S'.;8/3ij32iS'.aj^a2y-|-aj^&;8j^^2/3'^-"2'*y + ''2'^-/32/3/5i'^-°"iy = S.^l3,fi^ {yS.aaia,), by § 91 (3). 163.J It is evident, from these examples, that for special cases we can usually find modes of solution of the linear and vector equa- tion which are simpler in application than the general process of § 148. The real value of that process however consists partly in its enabling us to express inverse functions of 0, such as ((^+^)~^ for instance, in terms of direct operations, a property which will be of great use to us later ; partly in its leading us to the fundamental cubic ^^—m^^^ +mj(f>—m = 0, which is an immediate deduction from the equation of § 148, and whose interpretation is of the utmost importance with reference to the axes of surfaces of the second order, principal axes of inertia, the analysis of strains in a distorted solid, and various similar enquiries. 163.] When the fiinction <^ is its own conjugate, that is, when Spcpo' = Scrcfip for all values of p and o-, the vectors for which {<t>-ff)p = form in general a real and definite rectangular system. This, of course, may in particular cases degrade into one definite vector, and any pair of others perpendicular to it ; and cases may occur in which the equation is satisfied for every vector. 90 QUATERNIONS. [164, Suppose the roots oi mg= (§ 147) to be real and different, then VPi — ffiPi 1 ^j^ere p^, p^, P3 are three definite vectors determined Wi — y2P2 f jjy. ^jijg constants involved in <ji. <t>Pa =ffsP3' Hence ^i^2%P2 = '5-M#2 = S.pT,4>^P2, or = S.p^ip^pi, because ^ is its own conjugate. But (^^2 = fflPz) <l>^Pi=ffiPi> and therefore 9x9i^P-iPi = 9l^PiP2 = ^fi^f 1P2 > which, as g^ and g^ are by hypothesis different, requires SP\P2 = 0- Similarly 'S/'2P3 = 0, 'S'pgpj^ = 0. If two roots be equal, as g^, g^, we still have, by the above proof, iSpiPg = and Sp^p^ = 0. But there is nothing farther to determine />2 and P3, which are therefore any vectors perpendicular to py If all three roots be equal, every real vector satisfies the equation (<^_(/)p=0. 164.] Next, as to the reality of the three directions in this case. Suppose g^-'r^N — 1 to be a root, and let pg + tr^'v— 1 be the corresponding value of p, where g,^ and ^2 ai'c real numbers, pg and a^ real vectors, and v — 1 the old imaginary of algebra. Then ^{p^ + cTg a/— 1 ) = (^2 + >^2 ■v^^^) (P2 + "^a v'— ^). and this divides itself, as in algebra, into the two equations #2 = ^2^2 — '^2'^2) (/mJ-2= /?2P2+^2°'2- Operating on these by /S.o-g, /S.pg respectively, and subtracting the results, remembering our condition as to the nature of <^ <S'a-20P2 = Sp^^lT^, we have ^gC"'! +Pi) = 0. But, as o-g and p^ are both real vectors, the sum of their squares cannot vanish. Hence h^ vanishes, and with it the impossible part of the root. 165. J When ^ is self-conjugate, we have shewn that the equa- ^^^"^ g^— m^g^ + m-^g —m — Q has three real roots, in general different from one another. Hence the cubic in ^ may be written {.<i>-9i)i.^-9^{.4>-9s) = 0> 167.] SOLUTION OF EQUATIONS. 91 and in this form we can easily see the meaning of the cubic. For, let pi, p2, pg be three vectorg such that {^-ffi)pi = 0. {'t>—ff2)P2 = 0, {<t>—9^Ps = 0. Then any vector p may be expressed by the equation pS-PiP2Pa = pAP2P3P + P2.^-P3PiP + Pa^-PiP2P (§91). and we see that when the complex operation, denoted by the left- hand member of the above symbolic equation, is performed on p, the first of the three factors makes the term in pj vanish, the second and third those in p^ and pg respectively. In other words, by the successive performance upon a vector of the operations <f> — ^j, (p — ff^' ^—g^, it is deprived successively of its resolved parts in the direc- tions of Pi, p^, Pg respectively j and is thus necessarily reduced to zero, since pj, pg, pg are (because we have supposed g-^^, g^, g^ to be distinct) distinct and non-eoplanar vectors. 166.] If we take pj, pg, pg as rectangular K^zi^- vectors, we have — p = p-iSpjp + P2,8p2P + Ps'SpaP, whence # = —SiPx^pT^P—g^p^Sp^p—g^^Sp^p ; or, still more simply, putting i, j, h for p^, pg, pg, we find that any self-conjugate function may be thus expressed ^P = —9ii^P —ad^JP —9i^Skp, provided, of course, i, j, k be taken as roots of the equation Vp^p = 0. 167.] A very important transformation of the self-conjugate linear and vector function is easily derived from this form. We have seen that it involves three scalar constants only, viz. y^, g , g^. Let us enquire, then, whether it can be reduced to the fol- lowing form <j)p =/p + AF.{i + e/i:)p{i— eh), which also involves but three scalar constants/, h, e. Here, again, i, y, h are the roots of Vp^p = 0. Substituting for p the equivalent p = —iSip—jSjp—kSkp, expanding, and equating coefficients of i,j, k in the two expressions for <\>p, we find —g^ = —/+ ^^ (2 — 1 -[- e% -g,=-.f-k{2e'^ + l-e^). These give at once -(^1-^2) = 2-^, -{9z-9z) = Se^/J. 92 QUATERNIONS. [l68. Hence, as we suppose the transformation to be real, and therefore e^ to be positive, it is evident that ffi — ff^ and ^2 — ffa have the same sig^ ; so that we must choose as auxiliary vectors in the last term of <pp those two of the rectangular directions i, j\ k for which the coefficients g have the greatest and least values. We have then ^i^9j-h., 9i-9z ^=-\ {91-92), and f=\{gi+gi). 168.] We may, therefore, always determine definitely the vec- tors \, fi, and the scalar y, in the equation when <\> is self-conjugate, and the corresponding cubic has not equal roots, subject to the single restriction that is known, but not the separate tensors of X and fx. This result is important in the theory of surfaces of the second order, and will be considered in Chapter VII. 169.] Another important transformation of <^ when self-conju- gate is the following, ^p = aaVap + i^Sfip, where a and b are scalar s, and a and /3 unit-vectors. This, of course, involves sis scalar constants, and belongs to the most gen- eral form 4)p = —giPiSpiP—g2P2^P2p—9aP3^PaPy where pi, pg, p^ are the rectangular unit-vectors for which p and (pp are parallel. We merely mention this form in passing, as it be- Ipngs to the Jveal transformation of the equation of surfaces of the second order, which will not be farther alluded to in this work. It will be a good exercise for the student to determine a, ;8, a and b, in terms of i^^, yg. 93, ai"i Pi> P2, Pa- 170.] We cannot afford space for a detailed account of the sin- gular properties of these vector functions, and will therefore content ourselves with the enuntiation and proof of one or two of the most important. In the equation nKp'^FXpi. = F(\>'\<\)'ii. (§ 145), substitute \ for ^'K and ji for <^'p., and we have »«rqb'-i\<^'-V = ^FKix. Change ^ to <p+g, and therefore ^' to <\> +g, and m to %, we have a formula which will be found to be of considerable use. 1 72-] SOLUTION OF EQUATIOKS. 93 171.] Again, by § 147, Similarly -^ S.p (</> + ^)- V = t ■^P^" V + ^P + ^P^- Hence '^S.pi^+,)-^p-^S.pi^ + A)-^P = i,-,)[p^-'^]. That is, the functions are identiealj i. e. when equated to constants represent the same series of surfaces, not merely when g = h, but also, whatever be g and h, if they be scalar functions of p which satisfy the equation mS.p(j>-^p = gkp'^. This is a generalization, due to Hamilton, of a singular result ob- tained by the author *. 173.] The equations S.p((l>+g)-^p = 0,l S.p{<p + A)-^p=0,i ^ ' are equivalent to mSp(j)~^p+gSp\p+ff^p^ = 0, mSp<t>-'^p + hSpxp + h^p^ — 0. Hence m{\—x) Sp4>-''-p + {g—M) Spxp + {g^ — A^(c)p^ = 0, whatever scalar be represented by x. That is, the two equations (1) represent the same surface if this identity be satisfied. As particular cases let (1) aj=l, in which case Sp-\p+g + h = 0. (2) g—hx=0, in which case m{l - |)^p-i0-V + (/->^^f) = 0, or mSp~^4>~^P~9^ — "• (3) a- = |a. giving m{\- |-,) -Spr V + (^ - >^ fg) *PXP = 0, or m {A+g)Sp(l>-^p +g/iSpxp = 0. * Note on the CarteBian equation of the Wave-Surfaee. Quarterly Math. Jowmal, Oct. 1859. 94 QUATERNIONS. [l73- 173.] In various investigations we meet with the quaternion J = a<l>a + I3<j>p + y<l>y, where a, /3, y are three unit-vectors at right angles to each other. It admits of being put in a very simple form, which is occasionally of considerable importance. We have, obviously, by the properties of a rectangular unit- system ^ _ ^y^a + yatl>l3 + a^<}>y. As we have also s.afiy = _ i (§71(13)), a glance at the formulae of § 147 shews that at least if ^ be self-conjugate. Even if it be not, still (as will be shewn in § 1 74) ^p = ^'p + r^p, and the new term disappears in Sq. We have also, by § 90 (2), Vq=a{Sfi(l>y-Sy<pp)-i-P{Sy<j>a-Sa4r/) + y{Sa(l>^-Sl3<f>a) = a8fi{4)~<l/)y + fiSy{(t>-<tt)a + ySa{(l>—^')^ = aS.fiey + ^S.yea+ yS.aep = — {aSae + /S/S/Se + ySye) = e. [We may note in passing that this quaternion admits of being expressed in the remarkable form where V = OT-+S-7-+y-5-> ax ay dz and p=ax-\-^y-\-yz. We will recur to this towards the end of the work.] Many similar singular properties of <\> in connection with a rect- angular system might easily be given ; for instance, V{a F<l>^(j>y + j3 Ffycl)a + y F^a<p^) = mF(a<j)-^a + fi^'~^fi + y^'-'^y) = mF.Vi^'-'^p = 4>e ; which the reader may easily verify by a process similar to that just given, or (more directly) by the help of § 145 (4). A few others will be found among the Examples appended to this Chapter. 174.] To conclude, we may remark that as in many of the immediately preceding investigations we have supposed <f> to be self-conjugate, a very simple step enables us to pass from this to the non-conjugate form. For, if ^' be conjugate to (^, we have Sp(j>'(r = 8<T<pp, and also Spt^xr = Sa-^'p. 17 7-] soLUTioi^r OF equations. 95 Adding, we have SO that the function {<f> + <j)') is self-conjugate. Again, Sp(f>p = Spcj/p, which gives Sp{^—<^')p =. 0. Hence {<f>-~<l)')p = Fep, where, if ^ be not self-conjugate, e is some real vector, and therefore <t>P = \{<l> + <l>')p+\{4>-^')p =U<P + <t>')p+hrip. Thus every non-conjugate linear and vector function differs from a conjugate function solely iy a term of the form Fep. The geometric signification of this will be found in the Chapter on Kinematics. 175.] We have shewn, at some length, how a linear and vector equation containing an unknown vector is to be solved in the most general case ; and this, by § 138, shews how to find an unknown quaternion from any sufficiently general linear equation containing it. That such an equation may be sufficiently general it must have both scalar and vector parts : the first gives one, and the second three, scalar equations ; and these are required to determine com- pletely the four scalar elements of the unknown quaternion. 176.] Thus Tq = a being but one scalar equation, gives q = aJJr, where r is any quaternion whatever. Similarly Sq — a gives q — a +6, where d is any vector whatever. In each of these eases, only one scalar condition being given, the solution contains three scalar in- determinates. A similar remark applies to the following : Trq = a gives q = x + ad, and SUq = cos a, gives q = 006'^ , in each of which x is any scalar, and any unit vector. 177.] Again, the reader may easily prove that r.aVq^p, ^'^ QUATEENIONS. [178. where a is a given vector, gives, by putting Sq = x, Faq = p + cca. Hence, assuming Saq = y, we have aq=y + iDa + p, or ? = «+yo-i + a-^j8. Hercj the given equation being equivalent to two scalar con- ditions, the solution contains two scalar indeterminates. 178.] Next take the equation Faq = p. Operating by 8.a-\ we get Sq = 8a-^fi, so that the given equation becomes ra{Sa-^p+rq) = p, or FaFq = ^-aSa-^fi = aVa'^ From this, by § 168, we see that rq = a-^{x + aVa-^fi), whence q = Sa-i/3 + a-^ {« + a Fa-i/S) = a-i(/3 + a!), and, the given equation being equivalent to three scalar conditions, but one undetermined scalar remains in the value of q. This solution might have been obtained at once, since our equation gives merely the vector of the quaternion aq, and leaves its scalar undetermined. Hence, taking a; for the scalar, we have aq = Saq + Vaq 179.] Finally, of course^ from 0^ = 13, which is equivalent to four scalar equations, we obtain a definite value of the unknown quaternion in the form q = a-i^. 180.] Before taking leave of linear equations, we may mention that Hamilton has shewn how to solve any linear equation con- taining an unknown quaternion, by a process analogous to that which he employed to determine an unknown vector from a linear and vector equation j and to which a large part of this Chapter has been devoted. Besides the increased complexity, the peculiar fea- ture disclosed by this beautiful discovery is that the symbolic equation for a linear quaternion function, corresponding to the cubic 1 8 3- J SOLUTION OF EQUATIONS. 97 in (^ of § 162, is a biquadratic, so that the inverse function is .given in terms of the first, second, and third powers of the direct function. In an elementary work like the present the discussion of such a question would be out of place : although it is not very difficult to derive the more general result by an application of processes already explained. But it forms a curious example of the well-known fact that a biquadratic equation depends for its solution upon a cubic. The reader is therefore referred to the Mements of (Quaternions, p. 491. 181.] The solution of the following frequently-occurring par- ticular form of linear quaternion equation aq + qb = c, where a, b, and c are any given quaternions, has been effected by Hamilton by an ingenious process, which was applied in § 133 (5) above to a simple case. Multiply the whole bi/ Ka, and into b, and we have T'^a.q + Ka.qb=Ka.c, and aqb-\-qb'^ = cb. Adding, we have q {T^a + b'^ + 2Sa.b) = Ka.c + cb, from which q is at once found. To this form any equation such as a'qh'+c'qd' = e' can of course be reduced, by multiplication by c'-^ and into b'"'^. 183.] As another example^ let us find the differential of the cube root of a quaternion. If ^3 _ ,. we have q'^dq + qdq.q + dq.q^ = dr. Multiply by q, and into q~^, simultaneously, and we obtain q^dq.q~^ + q^dq + qdq.q = qdr.q-^. Subtracting this from the preceding equation we have dq.q^—q^dq.q~^= dr—qdr.q~^, or dq.q^—q^dq = dr.q—qdr, from which dq, or d{r^), can be found by the process of last section. The method here employed can be easily applied to find the differential of any root of a quaternion. 183.] To shew some of the characteristic peculiarities in the solution even of quaternion equations of the first degree when they are not sufficiently general, let us take the very simple one aq = qb, and give every step of the solution, as practice in transformations. H y** QUATERNIONS. [183. Apply Hamilton's process (§181), and we get T^a.q = Ka.gh, qh^ = aqb. These give q(THJrb'^-2bSa,) = 0, so that the equation gives no real finite value for q unless or b = Sa + l3TFa, where /3 is some unit-vector. By a similar process we may evidently shew that a = 8b + aTVb, a being another unit-vector. But, by the given equation, Ta = Tb, or S^a + T^ra = SH + TWb; from which, and the above values of «• and b, we sec that we may write So, Sb Wa = Wb=^' '^PP°''- If, then, we separate q into its scalar and vector parts, thus q = r + p, the given equation becomes {a. + a){r + p) = (r + p)(a + ^) (1) Multiplying out we have r{a—l3) = pfi — ap, which gives iS{a—p)p = 0, and therefore p = Fy{a—fi), where y is an undetermined vector. We have now r{a—p,) = p^-ap = ry{a-^).l3-aFy{a-p) = y{Safi+l)-{a-^)Spy + y{l+Sal3)-{a-fi)Say = -ia-l3)Sia + fi)y. Having thus determined r, we have q=-S{a + p)y+Fyia-p) 2q=-{a + p)y-y{a + fi) + y{a-p)-ia-p)y = —2ay—2y^. Here, of course, we may change the sign of y, and write the solution of aq = qb in the form q = ay + yfi, where y is any vector, and a = UFa, /3 = UFb. 185.] SOLUTION OF EQUATIONS. 99 To verify this solution, we' see by ( 1 ) that we require only to shew that aq =. qB. But their common value is evidently — y + ay/3. It will be excellent practice for the student to represent the terms of this equation by versor-arcsj as in § 54, and to deduce the above solution from the diagram directly. He will find that the solution may thus be obtained almost intuitively. 184.J No general method of solving quaternion equations of the second or higher degrees has yet been found ; in fact, as will be shewn immediatelyj even those of the second degree involve (in their most general form) algebraic equations of the sixteenth degree. Hence, in the few remaining sections of this Chapter we shall con- fine ourselves to one or two of the simple forms for the treatment of which a definite process has been devised. But firsts let us consider how many roots an equation of the second degree in an unknown quaternion must generally have. If we substitute for the quaternion the expression w-\rix-vjy + hz (§80), and treat the quaternion constants in the same way, we shall have (§ 80) four equations, generally of the second degree, to determine w, X, y, z. The number of roots will therefore be 2* or 16. And similar reasoning shews us that a quaternion equation of the mth degree has w* roots. It is easy to see, however, from some of the simple examples given above (§§ 175-178, &c.) that, unless the given equation is equivalent to four scalar equations, the roots will contain one or more indeterminate quantities. 185.] Hamilton has effected in a simple way the solution of the quadratic ^^ = qa-{- h, or the foUowingj which is virtually the same (as we see by taking the conjugate of each side), (f = aq + h. He puts q—\{a + w + p), where w is a scalar, and p a vector. Substituting this value in the first equation, we get a^ ^{iv + pf + 2wa + ap + pa = 2 {a^ -irWa-\- pa) + ^h, or (M; + /3)^ + i2p— pa = a^ + 4^. If we put Fa = a, S (a^ + 4b) = e, V{a^ + 45) = 2 y, this becomes {w + pY + 2Vap = c+2y; H a 100 QUATERNIONS. [l86. which, by equating separately the scalar and vector parts, may be broken up into the two equations 26)2 + p2 =: e, V[w-\-ci)p = y. The latter of these can be solved for p by the process of § 156, or more simply by operating at once by S.a which gives the value of S{w + a)p. If we substitute the resulting value of p in the former we obtain, as the reader may easily prove, the equation {w^-a^) (w*_cK>2 + y2)_(ray)2 = 0. The solution of this scalar cubic gives six values of w, for each of which we find a value of p, and thence a value of q. Hamilton shews {Lectures, p. 633) that only two of these values are real quaternions, the remaining four being biquaternions, and the other ten roots of the given equation being infinite. Hamilton farther remarks that the above process leads, as the reader may easily see, to the solution of the two simultaneous equations q + r = a, qr = -b; and he connects it also with the evaluation of certain continued fractions with quaternion constituents. (See the Miscellaneous Ex- amples at the end of the volume.) 186.] The equation q^ = aq+qb, though apparently of the second degree, is easily reduced to the first degree by multiplying 6y, and into, q~^, when it becomes l=q-^a + bq-'^, and may be treated by the process of § 181. 187.] The equation f' = aqb, where a and h are given quaternions, gives q{aqb) = {a,qb)q; and, by § 54, it is evident that the planes of q and aqh must coincide, A little consideration will shew that the solution depends upon drawing two arcs which shall intercept given arcs upon each of two great circles ; while one of them bisects the other, and is divided by it in the proportion oi m: 1. EXAMPLES TO CHAPTER V. 101 EXAMPLES TO CHAPTER V. 1. Solve the following equations: — (a.) V.apP = V.ay^. {h.) apfip = papj3. (c.) ap + pP = y. (d.) S.a^p + ^Sap — aVfip = y. (e.) p + ap^= afi. (/.) ap^p = p^pa. Do any of these impose any restriction on the generality of a and j3 ? 2. Suppose p = ix+Jy + iz, and (j)p = aiSip + hjSjp + ckSJcp ; put into Cartesian coordinates the following equations : — {a.) T4>p=l. (b.) Spil>^p=-l. (c.) S.p{<t>^-p^)-^p = -l. {d.) Tp = T4Up. 3. If X, p,, V be any three non-coplanar vectors, and q = F/xi'.(/)\+ FvX.(j)ix+ V\p,.(\>v, shew that q is necessarily divisible by S.\p,v. Also shew that the quotient is ^2-2 6, where Vep is the non-commutative part of <^p. Hamilton, Elements, p. 442. 4. Solve the simultaneous equations : — Sap =0,1 ^■> S.ap<bp = O.S S.ap<pp Sap =0,7 ^"•^ Sp<l>p =0.5 Sap =0,1 ^ ■' S.aipxp = 0. ) 5. If # = S/3/Sap+ Frp, where r is a given quaternion, shew that I = S {8.ay,a^asS.fisfi^^^) + ^S{r Fa^a^ . r/S^^i) + SrlS.apr - 2 (/Sar/S/Sr) + SrTr^ and m4>-^<T='2{ra^a^S.^^0^<T) + l.r.ar{r^<T.r)+ VarSr- VrStrr. Lectures, p. 561. 102 QUATERNIONS. 6. If [jog'] denote J>q~V'' {pqr) „ . S.plqr], to''] » {pqr) + lrq']Sp + lpr]Sq + \jip]Sr, and (i'?''*) !! '^'F L?***] > shew that the following relations exist among any five quaternions =jo{qrst) + q{rsip) + r{stpq) + s{tpqr) + t{pqrs), and q{prs() =-[rsf\Spq—[stp~\Srq + [tpr'\Ssq—[prs]Stq. Elements, p. 492. 7. Shew that if t^, i|f be any linear and vector functions, and a, /3, y rectangular unit-vectors, the vector e = V{4>a\lfa + ^fif^ + (j>r^y) is an invariant. [This will be immediately seen if we write it in the form 6 = F.<^V^p, which is independent of the directions of a, )3, y. But it is good practice to dispense with V.] If # = S^i^Cft and y^rp = ^rjiSC-yp, shew that this invariant may be expressed as -Sr#C or 2F7ji(^fi. Shew also that cfi-ijfp—yjfcpp = F6p. The scalar of the same quaternion is also an invariant, and may be written as —'S2^Sr]r]j^SC(i 8. Shew that if <^p = aSap + ^Sfip + ySyp, where a, ^, y are any three vectors, then -<t>-^pS^.afiy = aiSa,p + ^,S^^ + yj,SY^p, where a^ = Vfiy, &c. 9. Shew that any self-conjugate linear and vector function may in general be expressed in terms of two given ones, the expression involving terms of the second order. Shew also that we may write (^ +2; = a (OT-|-a;)2 + 5 (ct + a;) (oj +y) + c(<B +^)2, where a, I, c, x, y, z are scalars, and ct and co the two given func- tions. What character of generality is necessary in tn- and w ? How is the solution affected by non-self-conjugation in one or both ? 10. Solve the equations : — (a) q^ = Zqi+lOj. {b.) q^ =2q + i. (c.) qaq = bq + c. (d.) aq = qr = rb. EXAMPLES TO CHAPTER V. 103 11. Shew that ^FVcjyp = mVV(j)-'^p. 12. If (^ be self-conjugate, and a, /3, y a rectangular system, S.Fa(f>ari3(})^Fy(f>y = 0. 13. (f)\l/ and yj/cp give the same values of the invariants m, m^,m,^. 14. If <^' be conjugate to <^, <^<^' is self-conjugate. 1 5. Shew that ( Va&f + ( Y^fff + ( Yye)^ = 26^ if a, /3, y be rectangular unit-vectors. 16. Prove that V^ {<j)—ff)p = —pV^g+2,Vg. 17. Solve the equations : — '{a.) <^2 _ ^ . {b.) ^ + x = i^, I where one, or two, unknown linear and vector functions are given in terms of known ones. (Tait, Proc. JR. S. JE- 1870-71.) 18. If <^ be a self-conjugate linear and vector function, £ and rj two vectors, the two following equations are consequences one of the other, viz. : — f _ F.Tj^rj V ^ rm Si.rj<pri4>^ri /S^.f^^^^^' From either of them we obtain the equation This, taken along with one of the others, gives a singular theorem when translated into ordinary algebra. What property does it give of the surface S.p(fip(j)^p = 1 ? CHAPTER VI. GEOMETRY OP THE STRAIGHT LINE AND PLANE. 188.] Having, in the five preceding Chapters, given a brief exposition of the theory and properties of quaternions, we intend to devote the rest of the work to examples of their practical appli- cation, commencing, of course, with the simplest curve and surface, the straight line and the plane. In this and the remaining Chapters of the work a few of the earlier examples will be wrought out in their fullest detail, with a reference to the first five whenever a transformation occurs ; butj as each Chapter proceeds, superfluous steps will be gradually omitted, until in the later examples the full value of the quaternion processes is exhibited. 189.] Before proceeding to the proper business of the Chapter we make a digression in order to give a few instances of applications to ordinary plane geometry. These th-e student may multiply in- definitely with great ease. (a.) Euclid, I. 5. Let a and ^ be the vector sides of an iso- sceles triangle ; /3— a is the base, and Ta = T/3. The proposition will evidently be proved if we shew that a(a-^)-i=X/3(/3-a)-i (§ 52). This gives a(a-^)-i= (/3— a)-i/3, or (^— a)a = /3(a— j3), or _a2 = -/32. (5.) Euclid, I. 32. Let ABC be the triangle, and let u-= = r» AB 189.] GEOMETET OF STRAIGHT LINE AND PLANE. 105 where y is a unit-vector perpendicular to the plane of the triangle. If ^ = 1, the angle GAB is a right angle (§ 74). Hence 4=^^(§74). Let^ = m^, C=n'l- We have UlG=y'UAB, UCB = y''UCA, UBA=y"'UBC. Hence UBA=y'^.y''.y^UAB, or 1 = y+™+»>. That is l-\-m + n =2, or A + B+C=ii. This is, properly speaking, Legendre's proof ; and might have been given in a far shorter form than that above. In fact we have for any three vectors whatever, /3y a which contains Euclid's proposition as a mere particular case. (c.) Euclid, I. 35. Let y3 be the common vector-base of the parallelograms, a the conterminous vector-side of any one of them. For any other the vector-side is a + a?/3 (§ 28), and the proposition appears as Tn{a + xp) = TV^a (§§ 96, 98), which is obviously true. {d.) In the base of a triangle find the point from which lines, drawn parallel to the sides and limited by them, are equal. If a, j3 be the sides, any point in the base has the vector p = (1— ar)a+a;/3. For the required point which determines x. Hence the point lies on the line which bisects the vertical angle of the triangle. This is not the only solution, for we should have written T(l-a))Ta = Ti»!Tp, instead of the less general form above wMck tacitly assumes that 1—x and cc are positive. We leave this to the student. 106 QUATERNIONS. [iQO. (e.) If perpendiculars be erected outwards at the middle points of tlie sides of a triangle^ each being proportional to the corresponding sidcj the mean point of the triangle formed by their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equilateral. Let 2a, 2 /3j and 2 (a + y3) be the vector-sides of the triangle, i a unit-vector perpendicular to its plane, e the ratio in question. The vectors of the corners of the new triangle are (taking the corner opposite to 2/3 as origin) /Oj = a + eia, P2 = 2a + /3 + ei/3, P3 = a + /3— ei((a + /3). From these *(ft + P2 + /'3) = H4a+2;8) = k (2a-|-2(o + /3)), which proves the first part of the proposition. For the second part, we must have ^fe— Pi) = ^(P3— Pa) = 2'(pi— Pa). Substituting, expanding, and erasing terms common to all, the student wUl easily find 3 gS _ j _ Hence, if equilateral triangles be described on the sides of any tri- angle, their mean points form an equilateral triangle. 190.] Such applications of quaternions as those just made are of course legitimate, but they are not always profitable. In fact, when applied to plane problems, quaternions often degenerate into mere scalars, and become (§33) Cartesian coordinates of some kind, so that nothing is gained (though nothing is lost) by their use. Before leaving this class of questions we take, as an additional example, the investigation of some properties of the ellipse. 191.] We have already seen (§31 {h)) that the equation p = acos5 + /3sinfl represents an ellipse, Q being a scalar which may have any value. Hence, for the vector-tangent at the extremity of p we have Ap • ^ ^ „ OT = -^ = — asmd + i3cos0, do It which is easily seen to be the value of p when 6 is increased by - • Thus it appears that any two values of p, for which difiers by 1 94-] GEOMETRY OP STRAIGHT LINE AND PLANE. 107 IT' - , are conjugate diameters. The area of the parallelogram circum- scribed to the ellipse and touching it at the extremities of these diameters is, therefore, by § 96, ^TFp-^ = 4yr(acos0 + /3sin9)(— asine + /3eose) = 4yFa/3, a constant, as is well known. 193.] For equal conjugate diameters we must have y(aeose + /3sin0) = y(— a sin 9 4-/3 cos 0), or (a^— /3^) (cos^^— sin20) + 4^a/3cosesini9 = 0, a^ — B^ or tan 2 9 = „ „ '^ • The square of the common length of these diameters is of course a2 + ^^ , 2 because we see at once from § 191 that the sum of the squares of conjugate diameters is constant. 193.] The maximum or minimum of p is thus found ; dTp 1_ dp_ de ~~T^''dd' = — -^{ — (a^— 13^) cose sine + Sap icos^e—sm^0)). For a maximum or minimum this must vanish *, hence tan 2^= -5 — ^„, a^ — ^^ and therefore the longest and shortest diameters are equally inclined to each of the equal conjugate diameters. Hence, also, they are at right angles to each other. 194.] Suppose for a moment a and ;3 to be the greatest and least semidiameters. Then the equations of any two tangent-lines are p = a cos ^ + ;8 sin 5 + «(— a sin ^ + /3 cos 6), p = a cos ^1 H- j3 sin 0^ + Xy(^—a sin ^j -)- /3 cos 0-^. If these tangent-lines be at right angles to each other <?(— asin(9-|-/3cosfl)(— asin^i + /3cosei) = 0, or o? sin 6 sin 6^ -)- /3^ cos 5 cos ^j = 0. dB A little reflection will shew him that the latter equation involves an absurdity. The student must carefully notice that here we put -j— = 0, and not ^ = 0. civ 108 QUATEEJSriOKS. [195. Also, for their point of intersection we have, by comparing coeffi- cients of a, /3 in the above values of p, cos 6—xsmd = cos 6^ —x^ sin 6^ , sin O + x cos 6 = sin 6^ + x^ cos d-y . Determining x-y from these equations, we easily find the equation of a circle ; if we take account of the above relation between 6 and d^. Also, as the equations above give x = — x^, the tangents are equal multiples of the diameters parallel to them ; so that the line joining the points of contact is parallel to that joining the extremities of these diameters. 195.] Finally, when the tangents p = acosd +y3 sin5 +x (— asinfl +;Scos0), p = a cos 0^ + j8 sin d^ + x^ (—a sin 6^ + ^ cos 0^), meet in a given point p = aa + bp, we have a = cos 6 — x sin = cos 6^ — x^ sin 0^, h = sin 0-\-x cos = sin 0^ + x^ cos 0-^ . Hence x"^ = a^ + b^—1 = xl and a cos + b sin = 1 = a cos ^j + J sin 0^ determine the values of and x for the directions and lengths of the two tangents. The equation of the chord of contact is p = y{a cos 6 + /3 sin 6) + (1 —y) (a cos ^^ + /3 sin 0^). If this pass through the point p=jia + q^, we have ^ = ycos0 + (l— j^)cos9i, q = 2/svD.0 + {\ —y) sin 0^, from which, by the equations which determine and 5, , we get a])-irl(i=yJr\—y= 1. Thus if either a and h, or ^ and ^, be given, a linear relation con- nects the others. This, by § 30, gives all the ordinary properties of poles and polars. 196.] Although, in §§ 28-30, we have already g^ven some of the equations of the line and plane, these were adduced merely for their applications to anharmonic coordinates and transversals ; and not for investigations of a higher order. Now that we are prepared to determine the lengths and inclinations of lines we may investigate these and other similar forms anew. 200.] GEOMETRY OF STRAIGHT LINE AND PLANE. 109 197.] The equation of the indefinite line d/rawn through the origin 0, of which the vector OA, = a, forms apart, is evidently p = soa, or p II a, or Fap = 0, or Up =: Ua; the essential characteristic of these equations being that they are linear, and involve one indeterminate scalar in the value of p. We may put this perhaps more clearly if we take any two vectors, /3, y, which, along with a, form a non-coplanar system. Operating with S.Va^ and S.Vay upon any of the preceding equa- tions, we get S.afip = 0,1 and S.ayp = Q.\ *■ '' Separately, these are the equations of the planes containing a, /3, and o, y ; together, of course, they denote the line of intersection. 198.] Conversely, to solve equations (1), or to find p in terms of known quantities, we see that they may be written -S.pFa/3 = 0,-) S.pFay = 0,) so that p is perpendicular to Fa^ and Fay, and is therefore parallel to the vector of their product. That is, pII KFajSray, II -aS.a^y, or p = xa. 199.] By putting p— ;3 for p we change the origin to a point S where 0£ = — ^, or ^0 = y3 ; so that the equation of a line parallel to a, and passing through the extremity of a vector /3 drawn from the origin, is p—^ = xa, or p = p + xa. Of course any two parallel lines may be represented as p = /3 +xa, p = pj^+Xj^a; or Fa{p-fi) = 0, Fa{p-I3,) = 0. 200.] The equation of a line, drawn through the extremity of ^, and meeting a perpendicularly, is thus found. Suppose it to be parallel to y, its equation is p = ^ + xy. To determine y we know, first, that it is perpendicular to o, which gives Say = 0- 110 QUATERNIONS. [2OI. Secondly, o, ^, and y are in one plane, which gives S.a^y - 0. These two equations give y |{ r.aFaj3, whence we have p =: j3 + soa Vafi. This might have been obtained in many other ways ; for instance, we see at once that /3 = a- la/3 = a-^Safi + a-Wa^. This shews that ar^Va^ (which is evidently perpendicular to a) is coplanar with a and /3, and is therefore the direction of the re- quired line ; so that its equation is p = fi+ya-WaP, V the same as before if we put — ^-5- for x. la 201.J By means of the last investigation we see that —arWa^ is the vector perpendicular drawn from the extremity of /3 to the line p = xa. Changing the origin, we see that -a-ira(j3->/) is the vector perpendicular from the extremity of /3 upon the line p = y + xa. 203.] The vector joining £ (where OJS = fi) with any point in p =■ y + xa is y + Xa—p. Its length is least when dT{y+xa—0) = 0, or Sa{y + xa—^) = 0, i. e. when it is perpendicular to a. The last equation gives ■xa^+Sa{y-^) = 0, or xa=—a'' ^Sa{y — /3) . Hence the vector perpendicular is y-^-a-^Sa{y-0), or a-^Fa{y-fi)=-a-^Fa{l3—y), which agrees with the result of last section, 203.] To find the shortest vector distance between two lines p = fi + xa, and Pi=/?i + «'iai; 204.J GEOMETRY OF STRAIGHT LINE AND PLANE. Ill we must put dT{p—p^ =*0, or S{p-p^){dp-dp{) = 0, or S{p—pj){adx—aidxi) = 0. Since x and x^ are independent^ this breaks up into the two eon- 'litioiis Sa{p-p,)=0, Sajip-pj) = 0; proving the well-known truth that the required line is perpendicular to each of the given lines. Hence it is parallel to Faa-^, and therefore we have p—pi—l3 + xa—l3-^—x^ai = yFaai (1) Operate by S.aaj and we get This determines y, and the shortest distance required is [_N'ote. In the two last expressions T before S is inserted simply to ensure that the length be positive. If /S'.aai(/3 — /3j) be negative, then (§89) xS'.a^a(/3— ySj) is positive. If we omit the T, we must use in the text that one of these two ex- pressions which is positive.J To find the extremities of this shortest distance, we must operate on (1) with S.a and S.a^. We thus obtain two equations, which determine x and x^, as y is already known. A somewhat different mode of treating this problem will be dis- cussed presently. 204.] In a given- tetrahedron to find a set of rectangular coordinate axes, such that each axis shall ^ass through a pair of opposite edges. Let a, /3, y be three (vector) edges of the tetrahedron, one corner being the origin. Let p be the vector of the origin of the sought rectangular system, which may be called i, j, k (unknown vectors). The condition that i, drawn from p, intersects a is S.iap = (1) That it intersects the opposite edge, whose equation is 7^ = ^ + x{^-y), the condition is S.i(fi-y){p-^)=0, or Si{{^-y)p-M = (^ (2) There are two other equations hke (1), and two like (2), which can be at once written down. ^^^ QUATERNIONS. [205. Put p-y=a^, y-a = ^i, a-fi = y^, r^y = a^, Fya = /a^, Ta/S = y^, and the six become S.iap = 0, S.ia^p —Sia^ = 0, S.Jpp = 0, S.j0,p-8jp^ = 0, S.kyp = 0, S.hy-^p-Shy^ = 0. The two in i give i \\ aSa^-piSaa^ + Sarfi). Similarly, J\\^Sfi2p-p{Sfi^2 + SM> and i\\YSy2P-p{Syy^ + Sysp). The conditions of rectangulaffity, viz., SiJ = 0, SJi = 0, SM = 0, at once give three equations of the fourth order, the first of which is = Safi Sa^p S^^p-Sap Sa^{Sfifi^ + Sj3^p)-Sfip Sp^p{Saa^ + Sa^p) + p2 [Saa^ + Sa^p) {S^p^ + Sj3^). The required origin of the rectangular system is thus given as the intersection of three surfaces of the fourth order. 205.] The equation Sap = imposes on p the sole condition of being perpendicular to a ; and therefore, being satisfied by the vector drawn from the origin to any point in a plane through the origin and perpendicular to a, is the equation of that plane. To find this equation by a direct process similar to that usually employed in coordinate geometry, we may remark that, by § 29, we may write p = xj3 +yy, where /3 and y are any two vectors perpendicular to a. In this form the equation contains two indeterminates, and is often useful ; but it is more usual to eliminate them, which may be done at once by operating by Sm, when we obtain the equation first written. It may also be written, by eliminating one of the indeterminates only, as T^p = ya, where the form of the equation shews that Sa^ = 0. Similarly we see that Sa (p-/3) = represents a plane drawn through the extremity of ^ and perpen- dicular to a. This, of course, may, hke the last, be put into various equivalent forms. 306,] The line of intersection of the two planes 8.a (p-/3) = 0, ) and 5.ai(p-^i)=0,) ^ > 2o8.] GEOMETRY 0? STRAIGHT LINE AND FLAKE. 113 contains all points whose value of p satisfies both conditions. But we may write (§ 92), since a, a^, and Faa-^ are not coplanar, pS.aa-^Vaa-^^ — Vaa^SMa^p^ V.a-J^aai8ap+ F.F{aa^)aSa^p, or, by the given equations, —pT^ Vaa-^ = r.d^ Vaa^Sa^ + V. r{aa^ aSa^^ + x Yaa^, (2) where x, a scalar indeterminate, is put for S.aa^p which may have any value. In practice, however, the two definite given scalar equations are generally more useful than the partially indeterminate vector-form which we have derived from them. When both planes pass through the origin we have /3 = /S^ = 0, and obtain at once ^ ^ ^ jr^^ as the equation of the line of intersection. 207.] The plane passing through the origin, and through the line of intersection of the two planes (1), is easily seen to have the equation Sa^^iSap — SajSSaip = 0, or S{aSa^l3-^—a-^SaP)p = 0. For this is evidently the equation of a plane passing through the origin. And^ if p he such that Sap = Safi, we also have Saj^p = Sa^^^, which are equations (1). Hence we see that the vector aSa^^jSi — ajSaj3 is perpendicular to the vector-line of intersection (2) of the two planes (1), and to every vector joining the origin with a point in that line. The student may verify these statements as an exercise. 208.] To find the vector-perpendicular from the extremity of ^ on the plane Sap = 0, we must note that it is necessarily parallel to a, and hence that the value of p for its foot is p — ^^^a, where xa is the vector-perpendicular in question. Hence Sa {j3 + xa) = 0, which gives xa^:= —Sa^, or Xa = —a~^Sa/3. Similarly the vector-perpendicular from the extremity of /3 on the may easily be shewn to be -a-'^Sa(l3-y). I 114 QUATERNIONS. [209. 209.] The equation of the plane which passes through the ecctremities of a,^,y may be thus found. If p be the vector of any point in it) p—a, a—p, and /3— y lie in the plane, and therefore (§101) S.{p-a){a-^){fi-y)=0, or Sp{ra^-{- Vfiy+ rya)-S.a^y = 0. Hence, if 6 = a; ( F"a/3 + T/Sy + Fya) be the vector-perpendicular from the origin on the plane containing the extremities of a, y3, y, we have 6 = (ra/3+ r/3y+ Fyay^S.a^y. From this formula, whose interpretation is easy, many curious pro- perties of a tetrahedron may be deduced by the reader. Thus, for instance, if we take the tensor of each side, and remember the result of § 100, we see that T{ral3+rfiy+rya) is twice the area of the base of the tetrahedron. This may he more simply proved thus. The vector area of base is ir{d-fi) (y-/3) =-iiral3+ r^y+ Fya). Hence the sum of the vector areas of the faces of a tetrahedron, and therefore of any solid whatever, is zero. This is the hydrostatic proposition for solids immersed in a fluid subject to no external forces. 310.] Taking any two lines whose equations are p = 13 + xa, p =: jSj^ + X^Oi, we see that S.aaj(p — 6) ^ is the equation of a plane parallel to both. Which plane, of course, depends on the value of 8. Now if 8 = /3, the plane contains the first line ; if 8 = ^1, the second. Hence, liyVaa^ be the shortest vector distance between the lines, we have 5_„„^ {fi-^^-y Faa^) = 0, or TiyFaa^) = m(/3-^i) UFaa^, the result of § 203. 211.J Find the equation of the plane, passing through the origin, which makes equal angles with three given lines. Also find the angles in question. Let a, y3, y be unit-vectors in the directions of the lines, and let the equation of the plane be Sbp = 0. 2I3.J GEOMETRY OF STEAIGHT LINE AND PLANE. 115 Then we have evidently Sab = /S/38 = Syb = x, suppose, where ^ Tb is the sine of each of the required angles. But (§ 92) we have bS.a/Sy = X iFa^+ F^y+ Fya). Hence S.p ( VajS + V/3y + Fya) = is the required equation ; and the required sine is S.a^y ~ T{ra^+rpy+rya)' 312.] Find the locus of the middle points of a series of straight lines, each parallel to a given plane and having its extremities*in two fixed lines. Let 8yp — be the plane, and p = yg^a-^^ ^^ ^-y^x^a^, the fixed lines. Also let x and x-^ correspond to the extremities of one of the variable lines, is- being the vector of its middle point. Then, obviously, 2-a! = ^ + xa-\-^^+x-^a^. Also 8y{fi—^^->rXa—x^a^ = 0. This gives a linear relation between so and x-^ , so that, if we sub- stitute for Xj^ in the preceding equation, we obtain a result of the form ^^8+a;e, where 8 and e are known vectors. The required locus is, therefore, a straight line. 313.] Three planes meet in a point, and through the line of inter- section of each pai/r a plane is drawn perpendicular to the third ; prove that, in general, these planes pass through the same line. Let the point be taken as origin, and let the equations of the planes ^e Sap = 0, Sl3p = 0, Syp = 0. The line of intersection of the first two is || FajB, and therefore the normal to the first of the new planes is F.yFajB. Hence the equation of this plane is S.pF.yFa^ = 0, or SfipSay—SapS^y = 0, and those of the other two planes may be easily formed from this by cyclical permutation of a, /3, y. I a 116 QUATERNIONS. [214. We see at once that any two of these equations give the third by addition or subtraction, which is the proof of the theorem. 214.] Griven any number of points A, B, G, 8fc., whose vectors {from the origin) are a^, Og, a.^, 8fc.,find the plane through the origin for which the sum of the squares of the perpendiculars let fall upon it from these points is a maximum or minimum. Let ^wp = be the required equation, with the condition (evidently allowable) IW= 1. The perpendiculars are (§ 208) — ■nr~^*S'OTai, &c. Hence ^S^-^a is a maximum. This gives "^.SisaSadiTt! = ; and the condition that ■zn- is a unit-vector gives SnydvT = 0. Hence, as d-sr may have any of an infinite number of values, these equations cannot be consistent unless where a; is a scalar. The values of o are known, so that if we put ^ is a given self-conjugate linear and vector function, and therefore a; has three values {g^, g^, g^, § 164) which correspond to three mutually perpendicular values of -57. For one of these there is a maximum, for another a minimum, for the third a maximum- minimum, in the most general case when g^, g.^., g^ are all different. 215.] The following beautiful problem is due to MaccuUagh. Of a system of three rectangular vectors, passing through the origin, two lie on given planes, find the locus of the third. Let the rectangular vectors be ot, p, a. Then by the conditions of the problem gsyp = Spa = Sa^ = 0, and iSara- = 0, S^p = 0. The solution depends on the elimination of p and ot among these five equations. [This would, in general, be impossible, as p and ■in- between them involve six unknown scalars ; but, as the tensors are (by the very form of the equations) not involved, the five given equations are necessary and suflicient to eliminate the four unknown scalars which are really involved. Formally to complete the re- quisite number of equations we might write Ts^ = a, Tp = h, but a and h may have any values whatever.] EXAMPLES TO CHAPTER VI. 117 From Sasr = 0, /So-sr = 0, we have in- = xFaa: Similarly, from Sfip = 0, Sap = 0, we have P = y Vj3a: Substitute in the remaining equation S'srp = 0, and we have S.FaaF^a = 0, or Sa<rSj3<T — cr^Sa^ = 0, the required equation. As will be seen in next Chapter, this is a cone of the second order whose circular sections are perpendicular to a and /3. [The disappearance of x and y in the elimination in- structively illustrates the note above.J EXAMPLES TO CHAPTER VI. 1. What propositions of Euclid are proved by the mere /by»« of the equation p = {l—ai)a + xj3, which denotes the line joining any two points in space ? 2. Shew that the chord of contact, of tangents to a parabola which meet at right angles, passes through a fixed point. 3. Prove the chief properties of the circle (as in Euclid, III) from the equation p = a cos + ^ sin ; where Ta = Tfi, and Sa^ = 0. 4. What, locus is represented by the equation S^ap + p^= 0, where Ta= I? 5. What is the condition that the lines Fap = A Fa^P = ySi, intersect? If this is not satisfied, what is the shortest distance between them ? 6. Find the equation of the plane which contains the two parallel lines ra(p-/3)=0, Fa{p-^i) = 0. 7. Find the equation of the plane which contains ra(p-/3) = 0, and is perpendicular to gyp — o. 8. Find the equation of a straight line passing through a given point, and making a given angle with a given plane. Hence form the general equation of a right cone. 118 QUATERNIONS. 9. What conditions must be satisfied with regard to a number of given lines in space that it may be possible to draw through each of them a plane in such a way that these planes may intersect in a common line ? 10. Find the equation of the locus of a point the sum of the squares of whose distances from a number of given planes is con- stant. 11. Substitu^ "lines" for "planes" in (10). 12. Find the equation of the plane which bisects, at right angles, the shortest distance between two given lines. Find the locus of a point in this plane which is equidistant from the given lines. 1 3. Find the conditions that the simultaneous equations Sap = a, S^p = 6, Syp = c, may represent a line, and not a point. 1 4. What is represented by the equations {Sapf = {Sl3py^ = {Syp)^ where a, /3, y are any three vectors ? 15. Find the equation of the plane which passes through two given points and makes a given angle with a given plane. 16. Find the area of the triangle whose corners have the vectors a, /3, y. Hence form the equation of a circular cylinder whose axis and radius are given. 17. (Hamilton, Bishop Law's Fremium Ex., 1858). {a.) Assign some of the transformations of the expression /3— a' where a and /3 are the vectors of two given points A and B. {h.) The expression represents the vector y, or OC, of a point C in the straight line AB. (c.) Assign the position of this point C. 18. (Ibid.) (a.) If a, /3, y, 8 be the vectors of four points. A, B, C, B, what is the condition for those points being in one plane ? (h.) When these four vectors from one origin do not thus ter- minate upon one plane, what is the expression for the volume of the pyramid, of which the four points are the corners ? (c). Express the perpendicular S let fall from the origin on the plane ABC, in terms of a, y3, y. EXAMPLES TO CHAPTER VI. 119 19. Find the locus of a point equidistant from the three planes Sap = 0, S^p = 0, Syp = 0. 20. If three mutually perpendicular vectors be drawn from a point to a plane, the sum of the reciprocals of the squares of their lengths is independent of their directions. 21. Find the general form of the equation of a plane from the condition (which is to be assumed as a definition) that any two planes intersect in a single straight line. 22. Prove that the sum of the vector areas of the faces of any polyhedron is zero. CHAPTER VII. THE SPHERE AND CYCLIC CONE. 216.] Aftee that of the plane the equations next in order of simplicity are those of the sphere, and of the cone of the second order. To these we devote a short Chapter as a valuable prepara- tion for the study of surfaces of the second order in general. 217.] The equation y^ _ ^a or p^ = (^, denotes that the length of p is the same as that of a given vector a, and therefore belongs to a sphere of radius Ta whose centre is the origin. In § 107 several transformations of this equation were ob- tained, some of which we will repeat here with their interpretations. Thus ^(p + a)(p-a) = shews that the chords drawn from any point on the sphere to the extremities of a diameter (whose vectors are a and —a) are at right angles to each other. r(p + a)(p-a)= iTVap shews that the rectangle under these chords is four times the area of the triangle two of whose sides are a and p. (0 = (p + a)"^a(/3 + a) (see § 105) shews that the angle at the centre in any circle is double that at the circumference standing on the same arc. All these are easy consequences of the processes already explained for the interpretation of quaternion expressions. 218.] If the centre of the sphere be at the extremity of a the equation may be written T{p-a) = Tp, which is the most general form. If Ta = T/3, or a2 = /3^ 2 2 O.J THE SPHERE AND CYCLIC CONE. 121 in which ease the origin is a point on the surfaee of .the sphere, this becomes p^-2Sap = 0. From this, in the form Sp{p — 2a) = another proof that the angle in a semicircle is a right angle is de- rived at once. 219.] The converse problem is — Mnd the locus of tJiefeet of per- pendiculars let fall from a given point (p=/3) on planes passing through the origin. Let Sap = be one of the planes, then (§208) the vector-perpendicular is — a-^Saj3, and, for the locus of its foot, p = /3 — a-i/S'a/3, = orWap. [This is an example of a peculiar form in which quaternions some- times give us the equation of a surfaee. The equation is a vector one, or equivalent to three scalar equations ; but it involves the undetermined' vector a in such a way as to be equivalent to only- two indeterminates (as the tensor of a is evidently not involved). To put the equation in a more immediately interpretable form, a must be eliminated, and the remarks just made shew this to be possible.] Now {p-^Y =a-Wap, and (operating by S.fi) S^p-fi^=-a-Wafi. Adding these equations, we get P^-S^P = 0, so that, as is evident, the locus is the sphere of which y3 is a dia- meter. 220.] To find the intersection of the two spheres T(p-a) = h, and ^(p-«i) = ^/3i. square the equations, and subtract, and we have 2S{a-ai)p = a^-ai^-{^^-l3j^), which is the equation of a plane, perpendicular to a— aj the vector joining the centres of the spheres. This is always a real plane whether the spheres intersect or not. It is, in fact, what is called their Radical Plane. 122 QUATERNIONS. [221. 331.] Find the locus of a point the ratio of whose distances from two given points is constant. Let the given points be and A, the extremities of the vector a. Also let P be the required point in any of its positions, and OP=p. Then, at once, if n be the ratio of the lengths of the two lines, T{p-a) = nTp. This gives p^ — 2Sap + a^ = »2 p^, or, by an easy transformation, Thus the locus is a sphere whose radius is Tf- ^^> and whose centre is at JB, where 0£ = 5- > a definite point in the line OA. 1—n^ ^ 632.] ^in any line, OP, drawn from the origin to a given plane, OQ be taken such that OQ.OP is constant, fnd the locus of Q. Let Sap = a be the equation of the plane, ct a vector of the required surface. Then, by the conditions, T'HT Tp = constant = 5^ (suppose), and Z7«r = Up. From these p = -s= — = 5- • Substituting in the equation of the plane, we have aw^ + b^Saw = 0, which shews that the locus is a sphere, the origin being situated on it at the point farthest from the given plane. 333.] FiMd the locus of points the sum of the squares of whose dis- tances from a set of given points is a constant quantity. Find also the least value of this constant, and the corresponding locus. Let the vectors from the origin to the given points be oj, Oj, a„, and to the sought point p, then -c2 = {p-c^f+[p-a^f + + (p-a„)^ = np^-2Sp'2a+-S,{a^). Otherwise (,_^«/= _ flilli^!! + (?#, \ n' n n^ the equation of a sphere the vector of whose centre is — > i.e. whose centre is the mean of the system of given points. Suppose the origin to be placed at the mean point, the equation becomes /.2 j. y („i\ p2 ^ _^__+±S5l1 (for 2a = 0, § 31 (e)). 2 26. J THE SPHERE AND CYCLIC CONE. 123 The right-hand side is negative, and therefore the equation denotes a real surface, if ^2 ^ 2Ta^ as might have been expected. When these quantities are equal, the locus becomes a point, viz. the new origin, or the mean point of the system. 334.J If we differentiate the equation Tp = Ta we get Spdp — 0. Hence {^ \i7), p is normal ^ the surface at its extremity, a well- known property of the sphere. If tn- be any point in the plane which touches the sphere at the extremity of p, ta-— p is a line in the tangent plane, and therefore perpendicular to p. So that 8p{'7!-p) = 0, or S-arp = — Tp^ = a^ is the equation of the tangent plane. 225 .J If this plane pass through a given point B, whose vector is fi, we have ^^^ ^ „2. This is the equation of a plane, perpendicular to /3, and cutting from it a portion whose length is Tp ' If this plane pass through a fixed point whose vector is y we must have spy = a^ so that the locus of /8 is a plane. These results contain all the ordinary properties of poles and polars with regard to a sphere. 226.] A line drawn parallel to y, from the extremity of /3, has the equation p — a^^ This meets the sphere p2 _ ^2 in points for which w has the values given by the equation P^ + 2xSl3y-^x^y'^ = a^. The values of a; are imaginary, that is, there is no intersection, if The values are equal, or the line touches the sphere, if aV+^^/3y = 0, or S^l3y = y^P^-a^). This is the equation of a cone similar and similarly situated to the cone of tangent-lines drawn to the sphere, but its vertex is at the centre. That the equation represents a cone is obvious from the 124 QUATERNIONS. [227, fact that it is homogeneous in Ty, i.e. that it is independent of the length of the vector y. [It may be remarked that from the form of the above equation we see that, if x and x' be its roots, we have {xTy){x"I>y)=c?-fi\ which is Euclid, III, 35, 36, extended to a sphere.] 227.] Find the locus of the foot of the perpendicular let fall from a given point of a sphere on any tangent-plane. Taking the centre as origin, the equation of any tangent-plane may be written ^^p ^ „2_ The perpendicular must be parallel to p, so that, if we suppose it drawn from the extremity of a (which is a point on the sphere) we have as one value of ■or ■cT = a-\-xp. From these equations, with the help of that of the sphere we must eliminate p and x. We have by operating on the vector equation by S.'^ ■sr^ = SaiiT+xS'STp ■=■ /iSatsr + ara^. __ CT — a a^ (■or — a) Hence p = = — 5 — 5 Taking the tensors, we have (i!r2_^a^)2 = a2(ti^-a)^ the required equation. It may be put in the form and the interpretation of this gives at once a characteristic property of the surface formed by the rotation of the Cardioid about its axis of symmetry. 228.] We have seen that a sphere, referred to any point what- ever as origin, has the equation T{p-a) = T^. Hence, to find the rectangle under the segments of a chord drawn through any point, we may put p=xy; where y is any unit-vector whatever. This gives x^y^-2xSay+a^ = ^^, and the product of the two values of x is y 2 31.] THE SPHERE AND CYCLIC CONE. 125 .... • • This is positive, or the vector-chords are drawn in the same direc- tion, if T&<Ta, i.e. if the origin is outside the sphere. 229.] A, B are fixed points s and, leing the origin and P a point m space, jjp2 ^ ^pa = Qpa . find the locus ofP, and explain the result when LAOB is a right, or an obtuse, angle. Let OJ = a, 0B = ^,6P=p, then or p2_2^(a + y3)p=_(a2+/32), or y{p-(a4-/3)}=^/(-2&/3). While Sa^ is negative, that is, while LAOB is acute, the locus is a sphere whose centre has the vector o + /3. If ASa/3=0, or LAOB=-, the locus is reduced to the point p = a + /3. "If LAOB>- there is no point which satisfies the conditions. 230.] Bescriie a sphere, with its centre in a given line, so as to pass through a given point and touch a given plane. Let xa, where « is an undetermined scalar, be the vector of the centre, r the radius of the sphere^ /3 the vector of the given point, and Syp = a the equation of the given plane. The vector-perpendicular from the point xa on the given plane is (§208) {a-xSya)y-''. Hence, to determine x we have the equation T.{a-x8ya)y-'^ = T{xa-^) = r, so that there are, in general, two solutions. It will be a good exercise for the student to find from this equation the condition that there may be no solutioQj or two coincident ones. 231.] Bescribe a sphere whose centre is in a given line, and which passes through two given points. Let the vector of the centre be xa, as in last section, and let the vectors of the points be ^ and y. Then, at once, T{y-xa) =T{fi-xa) = r. Here there is but one sphere, except in the particular case when we have Ty = T^, and Say = Sa^, in which case there is an infinite number. 126 QUATERNIONS. [232. •' The student should carefiiUy compare the results of this section and the last, so as to discover why in general two solutions are possible in the one case, and only one in the other. 232.] A sphere touches each of two straight lines, which do not meet -. find, the locus of its centre. We may take the origin at the middle point of the shortest dis- tance (§203) between the given lines, and their equations will then be p = a-\-x^, where" we have, of course, Sa^ = 0, xSa/3i = 0. Let <r be the vector of the centre, p that of any point, of one of the spheres, and r its radius ; its equation is T{p-a) = r. Since the two given lines are tangents, the following equations in x and Xi must have pairs of equal roots, 2'(a4-«/8 — (7) = r, T{-a + a;^Pi-a-)=zr. The equality of the roots in each gives us the conditions S^I3<T =/32((a-(r)2+»-2), -S2/3i<T=^f((a + cr)2+r2). Eliminating r we obtain ^-^S^fia-fil^S^fi^a- = (a-o-)2-(a + <r)2 =-45a(7, which is the equation of the required locus. [As we have not, so far, entered on the consideration of the qua- ternion form of the equations of the various surfaces of the second order, we may translate this into Cartesian coordinates to find its meaning. If we take coordinate axes of so, y, z respectively parallel to |3, /3i, a, it becomes at once {x-\-myf^{jl-\-mxf' =^ pz, where m and p are constants ; and shews that the locus is a hy- perbolic paraboloid. Such transformations, which are exceedingly simple in all cases, will be of frequent use to the student who is proficient in Cartesian geometry, in the early stages of his study of quaternions. As he acquires a practical knowledge of the new calculus, the need of such assistance will gradually cease to be felt.J Simple as the above solution is, quaternions enable us to give one vastly simpler. For the problem may be thus stated — Find the locus of the point whose distances from two given, lines are equal. 2 34-] THE SPHEKE AND CYCLIC CONE. 127 And, with, the above notation, the equality of the perpendiculars is expressed (§ 201) by TV. (a -a)U^ = TV. (a + <t) U^^ , which is easily seen to be equivalent to the equation obtained above. 233.] Two spheres being given, shew that spheres which cut them at given atigles cut at right angles another fixed sphere. If be the distance between the centres of two spheres whose radii are a and i, the cosine of the angle of intersection is evidently ¥ab Hence, if a, a^, and p be the vectors of the centres, and «,«!,»• the radii, of the two fixed, and of one of the variable, spheres ; A and ^1 the angles of intersection, we have {p — af+a'^-\-r^= 2ar cos A, {p—aj)^ +al+r^ = 2ajrcosAj^. Eliminating the first power of r, we evidently must obtain a result sueh as (p— /S)^ + h^ + r^ = 0, where (by what precedes) /3 is the vector of the centre, and b the radius, of a fixed sphere {p-l3)^ + b^ = 0, which is cut at right angles by all the varying spheres. By effect- ing the elimination exactly we easily find b and y3 in terms of given quantities. 234.J To inscribe in a given sphere a closed polygon, plane or gauche, whose sides shall be parallel respectively to each of a series of given vectors. Let Tp = 1 be the sphere, a, fi, y , -q, 6 the vectors, n in number, and let Pi,P2, p„ , be the vector-radii drawn to the angles of the polygon. Then p2~Pi = ^i"' ^^-f ^^■ From this, by operating by S.{p2 + Pi), we get P2-Pi = = Sap2 + Sapi. Also = Vap2— Fapi. Adding, we get = apa + -^"Pi = "Pz + Pi «• Hence P2=— a~Vi"- [This might have been written down»at once from the result of §106.] Similarly p^ = — /3~V2/3 = ^"^ °-~^ Pi<^^> ^^• Thus, finally, since the polygon is closed, P»+i = Pi = i-T&'^rj-^ ^-''a-'p^a^ r,B. A 128 QUATERNIONS. [235. We may suppose the tensors of a, )3 t;, 6 to be each unity. Hence, if ^ ^ „^ ^g^ we have ffl-i = fl-i jj-i /3-1 a-\ which is a known quaternion ; and thus our condition becomes Pi = (-)"«">]«• This divides itself into two cases, according as n is an even or an odd number. If n be even, we have api = pya. Removing the common part p-^^Sa, we have Fp^Va = 0. This gives one determinate direction, ± Fa, for ^ ; and shews that there are two, and only two, solutions. If n be odd, we have ap^ = —p^a, which requires that we have Sa = 0, i. e. a must be a vector. Hence Sap^ = 0, and therefore pj^ may be drawn to any point in the great circle of the unit-sphere whose poles are on the vector a. 235.] To illustrate these results, let us take first the ease of m= 3. Here we must have S.aBy = or the three given vectors must" (as is obvious on other grounds) be parallel to one plane. Here afiy, which lies in this plane, is (§ 106) the vector-tangent at the first corner of each of the inscribed tri- angles; and is obviously perpendicular to the vector drawn from the centre to that corner. Ifn=4, we have p^ y f . ajSyb, as might have been at once seen from §106. 236.] Hamilton has given {Lectures, p. 674) an ingenious and simple process by which the above investigation is rendered ap- plicable to the more difficult problem in which each side of the inscribed polygon is to pass through a given point instead of being parallel to a given line. His process depends upon the integration of a linear equation in finite differences. By an immediate appli- cation of the linear and veetor function of Chapter V, the above solutions may be at once extended to any central surface of the second order. 237.] To find the equation of a cone of revolution, whose vertex is the origin. 240.J THE SPHERE AND CYCLIC CONE. 129 Suppose a, where jTa = l , to be its axis, and e the cosine of its semi-vertical angle ; then, if p be the vector of any point in the cone, SaUpz^^e, or S^ap = —e^p^. 238.] Change the origin to the point in the axis whose vector is xa, and the equation becomes { — X + SaTjrY ^—e^i^a + 'ury. Let the radius of the section of the cone made by Saur = retain a constant value &, while m changes ; this necessitates X Vb^ + m^ so that when x is infinite, e is unity. In this case the equation becomes ^2„^ ^. ^2 ^ j2 _ q^ which must therefore be the equation of a circular cylinder of radius b, whose axis is the vector a. To verify this we have only to notice that if w be the vector of a point of such a cylinder we must (§201) have TFaTu- = b, which is the same equation as that above. 239.] To find, generally, the equation of a cone which has a circular section : — Take the origin as vertex, and let the circular section be the intersection of the plane Sap = 1 with the sphere (passing through the origin) p2 = Sl3p. These equations may be written thus, SaUp= =-, -Tp = S^Up. Hence, eliminating Tp, we find the following equation which Up must satisfy— SaUpSfiUp =-l, or p^—SapS^p = 0, which is therefore the required equation of the cone. As a and /S are similarly involved, the mere form of this equation proves the existence of the subcontrary section discovered by Apol- lonius. 240.] The equation just obtained may be written S.UaUpS.Ul3Up = --^, 130 QUATERNIONS. [24 1. or, since a and y3 are perpendicular to the cyclic arcs (§ 59*), sinj» sinj!)'= constant, where j) and j)' are arcs drawn from any point of a spherical conic perpendicular to the cyclic arcs. This is a well-known property of such curves. 241 .J If we cut the cyclic cone by any plane passing through the origin, as gyp _ q^ then Fay and Ffiy are the traces on the cyclic planes, so that p = xUVay+yUF^y (§ 29). Substitute in the equation of the cone, and we get —x^—^^ + Pxy = 0, where P is a known scalar. Hence the values of x and _y are the same pair of numbers. This is a very elementary proof of the proposition in § 59*, that PL = MQ (in the last figure of that section). 243.] When x and ^ are equal, the transversal arc becomes a tangent to the spherical conic, and is evidently bisected at the point of contact. Here we have P=2 = 2S.UrayUrfiy+-^^-''^^'^' T.VayT^y This is the equation of the cone whose sides are perpendiculars (through the origin) to the planes which touch the cyclic cone, and from this property the same equation may readily be deduced. 243.] It may be well to observe that the property of the Stereo- graphic projection of the sphere, viz. that the projection of a circle is a circle, is an immediate' consequence of the above form of the equation of a cyclic cone. 244 J That § 239 gives the most general form of the equation of a cone of the second order, when the vertex is taken as origin, follows from the early results of next Chapter. For it is shewn in § 249 that the equation of a cone of the second order can always be put in the form 2 2.Sap8^p + Ap^ = 0. This may be written 8p<pP = 0, where <p is the self-conjugate linear and vector function (^/) = 2F.ap0 + (A + ^Safi)p. By § 168 this may be transformed to <i>P=pp+ F. Kpp., and the general equation of the cone becomes {j)-S\p.)p'^ + 2S\pSf^p = 0, which is the form obtained in § 239. 247-] THE SPHERE AND CYCLIC CONE. 131 • 245.] Taking the form Spct>p = as the simplest, we fiad by differentiation Sdp(f>p + Spd<pp = 0, or '2Sdp(j)p = 0. Hence (pp is perpendicular to the tangent-plane at the extremity of p. The equation of this plane is therefore (■nr being the vector of any point in it) Scj^p (t^-p) = 0, or, by the equation of the cone, aSct(^P = 0. 246.] T^e equation of the cone of normals to the tangent-planes of a given cone can he easily formed from that of the cone itself. For we may write it in the form S{<i>-^4,p)<pp = o, and if we put <pp-=a; a vector of the new cone, the equation becomes ■Sa4>-^<T = 0. Numerous curious properties of these connected cones, and of the corresponding spherical conies, follow at once from these equations. But we must leave them to the reader. 247.] As a final example, let vls find the equation of a cyclic cone when five of its vector-sides are given — i. e. find the cone of the second order whose vertex is the origin, and on whose surface lie the vectors a, A y, S, e. If we write = s.r{rapvbi)r(r^yrep)F{rybFpai (i) we have the equation of a cone whose vertex is the origin — ^for the equation is not altered by putting sep for p. Also it is the equation of a cone of the second degree, since p occurs only twice. Moreover the vectors a, ^,y, 6, e are sides of the cone, because if any one of them be put for p the equation is satisfied. Thus if we put /3 for p the equation becomes = s.v{rafirbe)r{rpyn^)r{rybr^a) = S.FiFa^ne) { F^aS.FyhF^yFe^- FybS.FfiaFPyFe^}. The first term vanishes because S.F{Fa^Fbe)Fl3a= 0, and the second because S.F^aF^yFflS = 0, since the three vectors FjSa, FjSy, Fej3, being each at right angles to /3, must be in one plane. As is remarked by Hamilton, this is a very simple proof of Pascal's K 2, 132 QUATERNIONS. Theorem — for (1) is the condition that the intersections of the planes of a, /3 and 8, e ; /3, y and e, p; y, 8 and p, a ; shall lie in one plane ; or, making the statement for any plane section of the cone, that the points of intersection of the three pairs of opposite sides, of a hexagon inscribed in a curve, may always lie in one straight line, the curve must he a conic section. EXAMPLES TO CHAPTER VII. 1 . On the vector of a point P in the plane Sap= 1 a point Q is taken, such that QO.OP is constant ; find the equation of the locus of Q. 2. "What spheres cut the loci of P and Q in (1) so that both lines of intersection lie on a cone whose vertex is ? 3. A sphere touches a fixed plane, and cuts a fixed sphere. If the point of contact with the plane be given, the plane of the inter- section of the spheres contains a fixed line. Find the locus of the centre of the variable sphere, if the plane of its intersection with the fiied sphere passes through a given point. 4. Find the radii of the spheres which touch, simultaneously, the four, given planes Sap = 0, Sj3p = 0, Syp = 0, Sbp = 1. [What is the volume of the tetrahedron enclosed by these planes ?] 5. If a moveable line, passing through the origin, make with any number of fixed lines angles 6, 6^, 02, &c., such that a cos.O + «! cos.^i + = constant, where «, «i, are constant scalars, the line describes a right cone. 6. Determine the conditions that Sp(j)p ^ may represent a ri^M cone. 7. What property of a cone (or of a spherical conic) is given directly by the following form of its equation, S.ipxp ^ ? 8. What are the conditions that the surfaces represented by Sp^p = 0, and S.ipKp = 0, may degenerate into pairs of planes ? EXAMPLES TO CHAPTER VII. 133 9. Find the locus of the vertices of all right cones which have a common ellipse as base. 10. Two right circular cones have their axes parallel, shew that the orthogonal projection of their curve of intersection on the plane containing their axes is a parabola. 11. Two spheres being given in magnitude and position, every sphere which intersects them in given angles will touch two other fixed spheres and cut a third at right angles. 12. If a sphere be placed on a tablcj the breadth" of the elliptic shadow formed by rays diverging from a fixed point is independent of the position of the sphere. 1 3. Form the equation of the cylinder which has a given circular section, and a given axis. Find the direction of the normal to the subcontrary section. 14. Given the base of a spherical triangle, and the product of the cosines of the sides, the locus of the vertex is a spherical conic, the poles of whose cyclic arcs are the extremities of the given base. 15. (Hamilton, Bishop Law's 'Premium Ex., 1858.) (a.) What property of a sphero-conic is most immediately in- dicated by the equation 5^5^=1? a p {b.) The equation {VKpf + {StipY = also represents a cone of the second order ; A. is a focal line, and jj. is perpendicular to the director-plane cor- responding. (c.) What property of a sphero-conic does the equation most immediately indicate ? 16. Shew that the areas of all triangles, bounded by a tangent to a spherical conic and the cyclic arcs, are equal. 17. Shew that the locus of a point, the sum of whose arcual dis- tances from two given points on a sphere is constant, is a spherical conic. 18. If two tangent planes be drawn to a cyclic cone, the four lines in which they intersect the cyclic planes are sides of a right cone. 19. Find the equation of the cone whose sides are the intersections of pairs of mutually perpendicular tangent planes to a given cyclic cone. 134 QUATERNIONS. 20. Find the condition that five given points may lie on a sphere. 21. What is the surface denoted by the equation where p = xa+y^ + zy, a, )3, y being given vectors, and x, y, z variable scalars ? Express the equation of the surface in terms of p, a, /3, y alone. 22. Find the equation of the cone whose sides bisect the angles between a fixed line and any line, in a given, plane, which meets the fixed line. What property of a spherical conic is most directly given by this result ? CHAPTER VIII. SURFACES OF THE SECOND OEDEB. 248.] The general scalar equation of the second order in a vector p must evidently contain a term independent of p, terms of the form S.apb involving p to the first degree, and others of the form S.aphpc involving p to the second degree^ a, h, c, &e. being constant quater- nions. Now the term S.apd may be written as SpF(da), or as S.{Sa+ ra)p{Sb+ Vb) = SaSpFb + SbSpFa + S-pFbra, each of which may evidently be put in the form Syp, where y is a known vector. Similarly * the term S.apbpc may be reduced to a set of terms, each of which has one of the forms Ap^, [Sapf, SapSpp, the second being merely a particular case of the third. Thus (the numerical factors 2 being introduced for convenience) we may write the general scalar equation of the second degree as follows : — 2S.SapS0p + Ap'^ + 2Syp = a (1) 249.] Change the origin to 1) where OJD = 6, then p becomes p + b, and the equation takes the form 22.SapS0p + Ap^+21(SapSpb + SfipSab) + 2AS&p+2Syp + 2-S.SabSl3b + Ab^ + 2Syb—C=0; from which the first power of p disappears, that is tie surface is referred to its centre, if 2(o-S'y38 + ;3<S'a8) + J5 + y = 0, (2) • For S.aphpc=S.capip=S.a'php = (2Sa'Sb—Sa'b)p' + 2Sa'p8bp; and in particular cases we may have Va'= Vb. 136 QUATERNIONS. [25O. a vector equation of the first degree^ wLicli in general gives a single definite value for 8, by the processes of Chapter V. [It would lead us beyond the limits of an elementary treatise to consider the special cases in which (2) represents a line, or a plane, any point of which is a centre of the surface. The processes to be employed in such special cases have been amply illustrated in the Chapter re- ferred to.] With this value of 6, and putting the equation becomes 2'L.SapSpp + Ap^=I). If 2? =^ 0, the surface is conical (a case treated in last Chapter) ; if not, it is an ellipsoid or hyperboloid. Unless expressly stated not to be, the surface will, when B is not zero, be considered an ellip- soid. By this we avoid for the time some rather delicate con- siderations. By dividing by B, and thus altering only the tensors of the constants, we see that the equation of central surfaces of the second order, referred to the centre, is (excluding cones) 2^{Sap8fip)+gp' = \ (3) 250.] Differentiating, we obtain 2'S{SadpSfip + SapS^dp} + 2gSpdp = 0, or 8.dp{1{a8pp + pSap) +gp} = 0, and therefore, by § 137, the tangent plane is <S(ot-p) {■2{cuS^p + pSap)+gp} = 0, i.e. S.'!!T{l(aSl3p + pSap)+ffp} = 1, by (3). Hence,if v = l{aSfip + pSap) + ffp, (4) the tangent plane is Svur = 1, and the surface itself is Si>p = 1. And, as v'^ (being perpendicular to the tangent plane, and satis- fying its equation) is evidently the vector-perpendicular from the origin on the tangent plane, v is called the vector ofpronmity. 251.] Hamilton uses for v, which is obviously a linear and vector function of p, the notation ^p, expressing a functional operation, as in Chapter V. But, for the sake of clearness, we will go over part of the ground again, especially for the benefit of students who have mastered only the more elementary parts of that Chapter. We have, then, (fip z=2{aSpp+^Sap)+ffp. 2 53-j SURFACES OF THE SECOND ORDEE. 137 With this definition of (f>, it is easy to see that («.) (j>{p + a-) = (f)p + <f>(T, &e., for any two or more vectors. (5.) (f) (a;/)) = :e(l>p, a particular case of (a), x being- a scalar, (c.) d(f>p = (l>{dp). {d.) Scr(^p = l,{SacTSfip + S^<TSap)+ffSp(T = Spcpa; or <p is, in this ease, self-conjugate. This last property is of great importance. 252.] Thus the general equation of central surfaces of the second degree (excluding cones) may now be written Sp4>P=l (1) Differentiating, Sdpipp + Spd(j>p = 0, which, by applying (c.) and then (d.) to the last term on the left, gives 2S^pdp=Q, and therefore, as in § 250, though now much more simply, the tangent plane at the extremity of p is 5(^-p)# = 0, or Stit^p := Sp(f>p = 1. If this pass through A{OA = a), we have Saipp = 1, or, by (d.), Spcfia = 1, for all possible points of contact. This is therefore the equation of the plane of contact of tangent planes drawn from J. 253.] To find the enveloping cone whose vertex is A, notice that {Sp4>p-l)+j){Sp4>a-lf = 0, where p is any scalar, is the equation of a surface of the second order touching the ellipsoid along its intersection with the plane. If this pass through A we have {Sa^a—\)-irp{Sa4,a.+ Vf = 0, and p is found. Then our equation becomes {Sp^p-l){Sa(j>a-l)—{Sp(j)a—lf = 0, (1) which is the cone required. To assure ourselves of this, transfer the origin to A, by putting p + a for p. The result is, using {a.) and (d.), {Sp(l)p+2Sp^a + Sa(j}a—l){Sa(j)a—l) — {Sp(pa + Safl}a-lf = 0, or Sptpp {Sacfia — 1 ) — (Sp(j)aY = 0, which is homogeneous in Tp^, and is therefore the equation of a cone. 138 QUATEENIONS. [254. Suppose A infinitely distant, then we may put in (1) xa for a, where x is infinitely great, and, omitting all but the higher terms, the equation of the cylinder formed by tangent lines parallel to a is {Sp<Pp—l)Sa<i>a—{8p^af = 0. 254.J To study the nature of the surface more closely, let us find the locus of the middle joints of a system of parallel chords. Let them be parallel to a, then, if ot be the vector of the middle point of one of them, ^a + xa and isr — xa are simultaneous values of p which ought to satisfy (1) of § 252. That is S.{'!!y±xa)i^{ts±xa)= \. Hence, by {a.) and {d.), as before. Surd's + x^Sa<j)a = 1, S'ST(l>a=zO (1) The latter equation shews that the locus of the extremity of ot, the middle point of a chord parallel to a, is a plane through the centre, whose normal is (pa ; that is, a plane parallel to the tangent plane at the point where OA cuts the surface. And {d.) shews that this relation is reciprocal — so that if /3 be any value of w, i. e. be any vector in the plane (1), a will be a vector in a diametral plane which bisects all chords parallel to /3. The equations of these planes are Sw^a = 0, S-ai^fi = 0, so that if F. ^a^/3 = y (suppose) is their line of intersection, we have Sycpa = = Sacj>y, \ Sy<t>^ = = Sfi,i>yA (2) and (1) gives Sficpa = = Sacp/B. ) Hence there is an infinite number of sets of three vectors a, /3, y, such that all chords parallel to any one are bisected by the diametral plane containing the other two. 255.] It is evident from § 23 that any vector may be expressed as a linear function of any three others not in the same plane, let then p = xa+yfi + zy, where, by last section, Sa^/3 = Sficpa = 0, Satpy = Sycpa = 0, Sl3(j)y = Sy<l>l3 = 0. And let Sacpa = 1. ) S/3ct,l3 = 1, [ Sycpy = 1, ) so that a, /3, and y are vector conjugate semi-diameters of the surface we are engaged on. 2 57-] SURFACES OF THE SECOND ORDEE. 139 Substituting the above value of p in the equation of the surface, and attending to the equations in a, /3, y and to (a.), {b.), and (cL), we have Sp<l)p = S{m + i/fi + zy) ^ {osa +yfi + zy), = x^ +y2 + z^ = 1 . To transform this equation to Cartesian coordinates, we notice that X is the ratio which the projection of p on a bears to a itself, &c. If therefore we take the conjugate diameters as axes of f, j;, f, and their lengths as a, b, c, the above equation becomes at once ^2 -I- §2 + g2 the ordinary equation of the ellipsoid referred to conjugate diameters. 256.] If we write —^^ instead of ^, these equations assume an interesting form. We take for granted, what we shall afterwards prove, that this halving or extracting the root of the vector func- tion is lawful, and that the new linear and vector function has the same properties («.), {b.), (c), {d.) (§ 251) as the old. The equation of the surface now becomes Sp^l,^p = -l, or ^^P^P = — 1) or, finally, T^p = 1. If we compare this with the equation of the unit-sphere Tp=l, we see at once the analogy between the two surfaces. TAe sphere can be changed into the ellipsoid, or vice versa, by a linear deformation of each vector, the operator being the function yjr or its inverse. See the Chapter on Kinematics. 257.] Equations (2) § 254 now become Sa\l/^I3= =S\j,a\j/^, &c., (1) so that yj/a, \lf^, \(ry, the vectors of the unit-sjahere which correspond to semi-conjugate diameters of the ellipsoid, form a rectangular system. We may remark here, that, as the equation of the ellipsoid referred to its principal axes is a case of § 255, we may now suppose i,j, and 3tj TJ Hy k to have these directions, and the equation is -^ + j^ -^ — 2 = ^j which, in quaternions, is {SipY {Sjpf {Skpf _ Sp<i>P=-^ + -^ + —^- - 1- We here tacitly assume the existence of such axes, but in all cases, by the help of Hamilton's method, developed in Chapter V, we at once arrive at the cubic equation which gives them. 140 QUATERNIONS. [258. It is evident from the last-written equation that iSip jSjp kSkp and a'' b'' c" ^ V a b ' which latter may be easily proved by shewing that And this expression enables us to verify the assertion of last section about the properties of ■^. As 8ip=. —X, &c., x,y, z being the Cartesian coordinates referred to the principal axes, we have now the means of at once transform- ing any quaternion result connected with the ellipsoid into the or- dinary one. 258.] Before proceeding to other forms of the equation of the ellipsoid, we may use those already given in solving a few problems. Find ike locus of a point when the perpendicular from the centre on its polar plane is of constant length. If OT be the vector of the point, the polar plane is Spt^T^ = 1, and the length of the perpendicular from is ^f- — (§ 208). Hence the required locus is T4>^ = G, or ^OT()!)V=-C2, a concentric ellipsoid^ with its axes in the same direction as those of the first. By § 257 its Cartesian equation is 259.] Find the locus of a point whose distance from a given point is always in a given ratio to its distance from a given line. Let p=xj3 be the given line, and A{OA=a) the given point, and let Safi = 0. Then for any one of the required points Tip-a) = eTrpp, a surface of the second order, which may be written p^-2Sap+a^ = e2 (6'2/3p_/3V)- Let the centre be at 8, and make it the origin, then p^ + 2Sp{b-a) + {b-af = e^S^.^{p + b)-fi^{p-\-by}, and, that the first power of p may disappear, {b-a) = e^{l3Sl3b-l3^), a linear equation for 6. To solve it, note that <Sa/3 = 0, operate by S.^ and we get (1 -e^/S^ + e^^^)S^b = S^b = 0. 2 6 1. J SURFACES OF THE SECOND ORDER. 141 Hence 8-a = -e^\ or Referred to this point as origin the equation becomes which shews that it belongs to a surface of revolution (of the second order) whose axis is parallel to /3, as its intersection with a plane S^p = a, perpendicular to that axis, lies also on the sphere P' e^a^ e^/3^a^ H-e2/32 {1 + e^^y In fact, if the point be the focus of any meridian section of an oblate spheroid, the line is the directrix of the same. 260.] A sphere, jiassing through the centre of an ellipsoid, is cut hy a series of spheres whose centres are on the ellipsoid and which pass through the centre thereof; find the envelop of the planes of inter- section. Let [p — df = o^ be the first sphere, i.e. p^ — 2Sap= 0. One of the others is p^ — 2&3-p = 0, where Snrcjyss- = 1 . The plane of intersection is S{7s — a)p = 0. Hence, for the envelop, (see next Chapter,) S'sr d>nr = 0, ) , , , „ , „ > where cr = afar, S'urp = 0, ) or <^OT = xp, {Vx = 0}, i.e. CT = co(l)~'^p. Hence x^Sp^-^.p =1, 1 and xSp<l)~'^p = Sap, ) and, eliminating x, Sp,j>-^p = {Sap)^ a cone of the second order. 261. J From a point in the outer of two concentric ellipsoids a tan* gent cone is d/rawn to the inner, find the envelop of the plane of contact. If Si!r(f>zT = 1 be the outer, and iSp^p = 1 be the inner, <f) and -^ being any two self-conjugate linear and vector functions, the plane of contact is Surxj/p = 1. . Hence, for the envelop, Sm'^p = 0, Sm\ tt'^P = 0, ) 3-'(^CT =: 0, ) 142 QUATERNIONS. [262. therefore (^ot = a!\//p, or tn- = x<^~^-\\ip. This gives xS.^p(^~'^^p = !> ) and x'^S.^p(\>~^-^p = 1, ) and therefore, eliminating x, S.^lrp,f>-^^jrp=-i, or S.p\j/tj)~^-^p = 1, another concentric ellipsoid, as \jf(l)~^\jf is a linear and vector func- tion = \ suppose ; so that the equation may be written Spxp= 1. 263.] Find the locus of intersection of tangent planes at the extre- mities of conjugate diameters. If a, /3, y be the vector semi-diameters, the planes are 8vr]f'^a= — \, •\ with the conditions § 257. Hence —^^v!S.-^w\i^^y=^'ss = ^a-\--<^^-V'^y, by § 92, therefore T^ts = vS, since yjra, ^jS, \jfy form a rectangular system of unit- vectors. This may also evidently be written fci/^^^ = - 3, shewing that the locus is similar and similarly situated to the given ellipsoid, but larger in the ratio -s/s : 1 . 263.] ' Find the locus of the intersection of three sjiheres whose dia- meters are semi-conjugate diameters of an ellipsoid. If a be one of the semi-conjugate diameters Sa\l/^a = — 1. And the corresponding sphere is p^—Sap=0, or p^—S\^ai^~^p = 0, with similar equations in /3 and y. Hence, by § 92, y}f-^pS.\jra\j/^\j/y = -i'-'^p = p'^{\lfa + \l/l3 + \l/y), and, taking tensors, T^'^p = VsTp^, or ^-^"^=^3, or, finally, Sprj/'^p ;^-3p\ This is Fresnel's Surface of Elasticity in the Undulatory Theory. 264.] Before going farther we may prove some useful properties of the function ^ in the form we are at present using — viz. iSip jSjp kSkp 265.] SURFACES OP THE SECOND OKDER. 143 We have p = and it is evident that (jii =■ ^J = -i # = -^2' Hence _ iSip jSjp kSkp <^V=-^- a* b* C <j>~^P = aHSip + bySjp + c^/cSkp, Also and so on. Again, if a, /3, y be any rectangular unit-vectors But as we have Again, „2 ^ ^2 &c. = &c. (Sipf + {Sjp)^ + {Skpf=-p\ Sa(f,a + Sfi<l>^ + Sy<py = 1^ + ^+1^ S. ASia iSi^ ASiy *.♦,*,=«.(= + ...)(5^ + ...)('2?+...) Sia Siy ,2 b^ '■ b^ ■ Sjy Ska c2 Sk^ c2 Sky — 1 a^b^c' Sia, Sja, Ska Sip, Sjl3, Skp Siy, Sjy, Sky = + a^b^c^ And so on. These elementary investigations are given here for the benefit of those who have not read Chapter V. The student may easily obtain all such results in a far more simple manner by means of the formulae of that Chapter. 265.] MnAthe locus of intersection of a rectangular system of three tangents to an ellipsoid. If tn- be the vector of the point of intersection, a, /3, y the tangents, then, since •m + xa should give equal values of a; when substituted in the equation of the surface, giving S {m + Xa) <p {-or + xa) = 1, or x^Sa(\)a + 2xS^(f>a + (/Soti^ct — 1 ) = 0, we have {S^ipaY = Sa<l)a {S-sr(j)w—l). Adding this to the two similar equations in /3 and y (/Sa^ti7)2 + (S/Scp^f + {Sy(l)wf = {Sa^a + <S/3<^/3 + Sy<l>y) (/Stsrc/.w - 1 ), 144 QUATERNIONS. [266. or -{<}>^f = (1 + 1, + ^) {S:^^-l), an ellipsoid concentric with the first. 366.] If a rectangular system of chords he drawn through any point within an ellipsoid, the sum of the reciprocals of the rectangles under the segments into which they are divided is constant. With the notation of the solution of the preceding problem, w giving the intersection of the vectors, it is evident that the product of the values of x is one of the rectangles in question taken nega- tively. Hence the required sum is 1 £ 1 Sta^TH — 1 Ssy^'ST — 1 This evidently depends on Smcfrar only and not on the particular directions of a, ^,y : and is therefore unaltered if ■nr be the vector of any point of an ellipsoid similar, and similarly situated, to the given one. [The expression is interpretable even if the point be exterior to the ellipsoid.] 267.] Shew that if any rectangular system of three vectors he drawn from a point of am, ellipsoid, the plane containing their other extremities passes through a jimed point. Find the locus of the latter point as tlie former varies. With the same notation as before, we have SsT(j)Zl7 ^ 1, and 8 (;sr + X a) (j) (tn- + xa) = 1 ; , , „ 2Sa<b-sr thereiore x = • ocupa Hence the required plane passes through the extremity of Sa(pa and those of two other vectors similarly determined. It therefore passes through the point whose vector is aSa^Tjy + ^SlB^yar + ySycjiZT Sa(t>a + S^(l)l3 + 8y(l>y ' or 6 = ^+^-^ (§173). Thus the first part of the proposition is proved. 268.] SURFACES OF THE SECOND ORDER. 145 OTo -1 But we have also ot = — ("(^ + — ) whence by the equation of the ellipsoid we obtain the equation of a concentric ellipsoid. 268.] Find the directions of the three vectors which are parallel to a set of conjugate diameters iti each qf two central surfaces of the second degree. Transferring the centres of both to the origin, let their equations be Sp(t>p.— 1 or 0,; and Sp\l/p= 1 or O.S ^' If a, l3, y be vectors in the required directions, we must have (§254) Sa(p^ = 0, Sa\{/^ = 0, \ S^<t>y^Q, S^^lry=Q^ (2) Sy(l>a = 0, Syfa = 0. ) From these equations 0a || V^y || ^a, &c. Hence the three required directions are the roots of r.<t>pi'p = o (3) This is evident on other grounds, for it means that if one of the surfaces expand or contract uniformly till it meets the other, it will touch it successively at points on the three sought vectors. We may put (3) in either of the following forms — F.p4,-^irp=0,]^ or r.p\/f-i(|)p= 0;i ^ '' and, as <j) and v/f are given functions, we find the solutions by the processes of Chapter V. [iVbfe. As (j)~^^ and V~^^ ^^^ ^°^> ^^ general, self-conjugate functions, equations (4) do not signify that a, /3, y are vectors parallel to the principal axes of the surfaces <S.p0-Vp = 1> S.p^jf-^(t)p = 1. In these equations it does not matter whether (j)~^^ is self-conjugate or not ; but it does most particularly matter when they are differ- entiated, so as to find axes, &c.] Given two surfaces of the second degree, there exists in general a set of Cartesian axes, whose directions are those of conjugate diameters in every one of the surfaces of the second degree passing through the inter- section of the two surfaces given. L 146 QUATERNIONS. [269. For any surface through the intersection of Sp(j)p=l and S{p—a)^{p—a) = e, is fSp4>p—8{p—a)-f{p — a)=f—e, where/ and e are scalars. The axes of this depend only on the term Sp{fct>-y},)p. Hence the set of conjugate diameters which are the same in all are the roots of J'i/'t>-^)pU'i4>-^)p=0, or rcpp^p=0, as we might have seen without analysis. The locus of the centres is given by the equation ('/'-/^)P-V'« = o> where/" is a scalar variable. 269.] Find the equation of the ellipsoid of which three conjugate semi-diameters are given. Let the vector semi-diameters be a, j3, 7, and let 8p4,p = 1 be the equation of the ellipsoid. Then (§ 255) we have Sa(i)a = \, Sa(pfi = 0, Sycjyy = 1, Sy^a ^ ; the six scalar conditions requisite (§ 139) for the determination of the linear and vector function (j). They give a \\ V(j)^(j)y, or xa = (j}~^ F/3y. Hence ■ cc = ccScupa = S.afiy, and similarly for the other combinations. Thus, as we have pS.a^y = cuS.^yp+^S.yap + yS.afip, we find at once <j)pS^.al3y = Fl3yS.j3yp + VyaB.yap+ FafiS.a^p; and the required equation may be put in the form S^.afiy = S^.a^p + S^.fiyp + S^.yap. The immediate interpretation is that if four tetrahedra be formed iy growping, three and three, a set of semi-conjugate vector axes of an ellipsoid and any other vector of the surface, the sum of the squares of the volumes of three of these tetrahedra is equal to the square of the volume of the fourth. 2 7 2. J SURFACES OP THE SECOND ORDER. 147 • 370.] When the equation of a surface of the second order can be put in the form Sp(()-''-p = I, (1) where (<^-^)(<#>-^i)(<l'-^2) = 0. _ we know that ff, ff^ , g^ are the squares of the principal semi-diameters. Hence, if we put (|) + ^ for <^ we have a second surface, the diifer- enees of the squares of whose principal semiaxes are the same as for thefirst. Thatis, 8p{<^ + h)-'^p=\ (2) is a surface confocal with (1). From this simple modification of the equation all the properties of a series of confocal surfaces may easily be deduced. We give one as an example. 271.] Any two confocal surfaces of the second order, wJiich meet, intersect at right angles. For the normal to (2) is, evidently, and that to another of the series, if it passes through the common point whose vector is p, is there (<^ + /ii)-V. But ^.(<^+^)-v(^+^o-P = ^•P (^^,)(^^^y and this evidently vanishes if h, and h-^ are different, as they must be unless the surfaces are identical. 272.] To find the conditions of similarity of two central surfaces of the second order. Referring them to their centres, let their equations be 8p<^'p=\.\ ^'^ Now the obvious conditions are that the axes of the one are pro- portional to those of the other. Hence, if g^-\-m^g'^ + m^g ^m=fi,\ ^g'nm\g'+m'=0,i ^'^ /' + be the equations for determining the squares of the reciprocals of the semiaxes, we must have —^=IJ; -^ = IJ.^, — = IJ,^, (3) m^ m^ m ' where \x. is an undetermined scalar. Thus it appears that there are but two scalar conditions necessary. Eliminating jn we have ni'\ _ nn'y m'm\ _ m'\ m% ~ %' mm^ ~ mf ^ ■' which are equivalent to the ordinary conditions. L a 148 QUATERNIONS. [273. 273.] Find. the greatest and least semi-diameters of a central plane sectioti of an ellipsoid. Here Spcl,p = I I Sap=o] ^ > together represent the elliptic section ; and our additional condition is that Tp is a maximum or minimum. Differentiating the equations of the ellipse, we have S(f>pdp = 0, Sadp = 0, and the maximum condition gives dTp = 0, or Spdp = 0. Eliminating the indeterminate vector dp we have S.apcf>p = (2) This shews that tAe maximum or minimum' vector, the normal at its extremity, and the perpendicular to the plane of section, lie in one plane. It also shews that there are but .two vector-directions which satisfy the conditions, and that they are perpendicular to each other, for (2) is satisfied if ap be substituted for p. We have now to solve the three equations (1) and (2), to find the vectors of the two (four) points in which the ellipse (1) intersects the cone (2). We obtain at once 4>p = xV.<^~'^dVap. Operating by S.p we have 1 = xp^Sa(l)~^a. XT 2 J. Sp(j)-''-a Hence p'op = p-a „ , ^ »' '=^('-''«--> « fromwhich ■ S.a{l—p^(f))-^a= ; (4) a quadratic equation in p^, from which the lengths of the maximum and minimum vectors are to be determined. By § 147 it may be written mp*Sa(l)-'^a—p^S.a{m2—(t>)a+a' = (5) [If we had operated' by 8.<p-^a or by 8.(pr^p, instead of by S.p, we should have obtained an equation apparently different from this, but easily reducible to it. To prove their identity is a good exercise for the student.] Substituting the values of p^ given by (5) in (3) we obtain the vectors of the required diameters. [The student may easily prove directly that {\—pl<f>)-'^a and {l—pl^)-^a 276.] SURFACES OF THE SECOND ORDER. 149 are necessarily perpendicular to each other, if both be perpendicular to a, and if pf and p| be different. See § 271.] 274.] By (5) of last section we see that 2 2 _ "^ Hence the area of the ellipse (1) is V — mSa<f)~^a Also the locus of normals to all diametral sections of an ellipsoid, whose areas are equal, is the cone Sa(t>-'^a = Co?. When the roots of (5) are equal, i.e. when {m..fl^—Sa^af = ima'^Satp-'^a, (6) the section is a circle. It is not difficult to prove that this equation is satisfied by only two "Values of Ua, but another quaternion form of the equation gives the solution of this and similar problems by inspection. (See § 275 below.) 275.] By § 168 we may write the equation Sp<f>p =: 1 in the new form S.Kpfxp + pp^ = 1, where ^ is a known scalar, and A. and f/. are definitely known (with the exception of their tensors, whose product alone is given) in terms of the constants involved in </>. [The reader is referred again also to §§ 121, 122.] This may be written 2SkpSij.p + {p—SKiJ.)p^ = l (1) From this form it is obvious that the surface is cut by any plane perpendicular to A. or fi in a circle. For, if we put S\p = a, we have 2aSixp + {p—S\ix)p^ = 1, the equation of a sphere which passes through the plane curve of intersection. Hence X and n o( § 168 are the values of a in equation (6) of the preceding section. 276.] Any two circular sections of a central surface of the second order, whose planes are not parallel, lie on a sphere. For the equation {S\p—a) (Sixp — b) = 0, where a and b are any scalai* constants whatever, is that of a system of two non-parallel planes, cutting the surface in circles. Eliminating the product SKpS^p between this and equation (1) of last section, there remains the equation of a sphere. 150 QUATEENIONS. [277. 277.] To find the generating lines of a central surface of the second order. Let the equation be Spcpp = 1 ; then, if a be the vector of any point on the surface, and ■nr a vector parallel to a generating line, we must have p = a + xm for all values of the scalar x. Hence 8 {a + xw) <^ (o + xm) = 1 , which gives the two equations ■=o.J The first is the equation of a plane through the origin parallel to the tangent plane at the extremity of a, the second is the equation of the asymptotic cone. The generating lines are therefore parallel to the intersections of these two surfaces, as is well known. Froni these equations w.e have ycfysT = Fota- where _^ is a scalar to be determined. Operating on this by S.^ and S.y, where y3 and y are any two vectors not coplanar with a, we have S^{ycl>^+ra^) = 0, Sm{i/<t,y—rya) = (1) Hence S.<})a (j^^^ + Fa/3) {y(j)y— Vya) = 0, or my^S.a^y—SacpaS.a^y = 0. Thus we have the two values Sa<f>'sr = 0, a /I belonging to the two generating lines. 278.] But by equation (1) we have zm = r.(y^/3+ Va^) {y^y— Vyd) = my"^ (j)-^ V^y + yV.^a V^y — aS.aVfiy ; which, according to the sign of y, gives one or other generating line. Here V^y may be any vector whatever, provided it is not per- pendicular to a (a condition assumed in last section), and we may write for it 6. Substituting the value of y before found, we have zvT = (t)-^d—ajSa0 + ^ — Fd>a0, 278.J SURFACES OF THE SECOND ORDER. 151 or, as we may evidently write it, = <i>-'^{r.ar4>ae)±J~r^ae (2) Put r = V^a6, and we have zur = d>-^ Far + ^— t, ~ ^ m with the condition Srcpa = 0. [Any one of these sets of values forms the complete solution of the problem ; but more than one have been given, on account of their singular nature and the many properties of surfaces of the second order which immediately follow from them. It will be excellent practice for the student to shew that is an invariant. This may most easily be done by proving that V.y^e-^Oi = identically.] Perhaps, however, it is simpler to write a for F/3y, and we thus «CT- = — d) '■ya yaAa + x/ — Va<i>a. ^ m [The reader need hardly be reminded that we are dealing with the general equation of the central surfaces of the second order — the centre being origin.] EXAMPLES TO CHAPTER VIII. 1 . Find the locus of points on the surface Sp<^p = 1 where the generating lines are at right angles to one another. 2. Find the equation of the surface described by a straight line which revolves about an axis, which it does not meet, but with which it is rigidly connected. 3. Find the conditions that Sp^p = 1 may be a surface of revolution, with axis parallel to a given vector. 4. Find the equations of the right cylinders which circumscribe a given ellipsoid. 5. Find the equation of the locus of the extremities of perpen- diculars to central plane sections of an ellipsoid, erected at the 152 QUATERNIONS. centre, their lengths being the principal semi-axes of the sections. [Fresnel's Wave-Surface. See Chap. XI.] 6. The cone touching central plane sections of an ellipsoid, which are of equal area, is asymptotic to a confocal hyperboloid. 7. Find the envelop of all non-central plane sections of an ellip- soid whose area is constant. 8. Find the locus of the intersection of three planes, perpendicular to each other, and touching, respectively, each of three confocal surfaces of the second order. 9. Find the locus of the foot of the perpendicular from the centre of an ellipsoid upon the plane passing through the extremities of a set of conjugate diameters. 10. Find the points in an ellipsoid where the inclination of the normal to the radius-vector is greatest. 1 1 . If four similar and similarly situated surfaces of the second order intersect, the planes of intersection of each pair pass through a common point. 12. If a parallelepiped be inscribed in a central surface of the second degree its edges are parallel to a system of conjugate dia- meters. 13. Shew that there is an infinite number of sets of axes for which the Cartesian equation of an ellipsoid becomes - x^-^y'^+z^ = e^. 14. Find the equation of the surface of the second order which circumscribes a given tetrahedron so that the tangent plane at each angular point is parallel to the opposite face; and shew that its centre is the mean point of the tetrahedron. 15. Two similar and similarly situated surfaces of the second order intersect in a plane curve, whose plane is conjugate to the vector joining their centres. 16. Find the locus of all points on Sp(i>p = 1, where the normals meet the normal at a given point. Also the locus of points on the surface, the normals at which meet a given line in space. 17. Normals drawn at points situated on a generating line are parallel to a fixed plane. 18. Find the envelop of the planes of contact of tangent planes drawn to an ellipsoid from points of a concentric sphere. Find the locus of the point from which the tangent planes are drawn if the envelop of the planes of contact is a sphere. EXAMPLES TO CHAPTER VIII. 153 19. The sum of the reciprocals of the squares of the perpendiculars froiQ the centre upon three conjugate tangent planes is constant. 20. Cones are drawn, touching an ellipsoid, from any two points of a similar, similarly situated, and concentric ellipsoid. Shew that they intersect in two plane curves. Find the locus of the vertices of the cones when these plane sec- tions are at right angles to one another. 2 1 . Find the locus of the points of contact of tangent planes which are equidistant from the centre of a surface of the second order. 22. From a fixed point A, on the surface of a given sphere, draw any chord AB; let 1/ be the second point of intersection of the sphere with the secant £D drawn from any point £ ; and take a radius vector AE, equal in length to SB', and in direction either coincident with, or opposite to, the chord AD : the locus of S is an ellipsoid, whose centre is A, and which passes through B. (Hamilton, Elements, p. 227.) 23. Shew that the equation p (e2_ 1) (e + Saa) = (Sapf - 2eSapSa'p + (Sa'pf + (1 -e^) p\ where e is a variable (scalar) parameter, and a, a' unit- vectors, repre- sents a system of eonfocal surfaces. {Ibid. p. 644.) 24. Shew that the locus of the diameters of Sp<pp = 1 which are parallel to the chords bisected by the tangent planes to the cone Spfp = is the cone S.p(f>yjf~''-'(f)p = 0. 25. Find the equation of a cone, whose vertex is one summit of a given tetrahedron, and which passes through the circle circum- scribing the opposite side. 26. Shew that the locus of points on the surface Sp<f)p = 1, the normals at which meet that drawn at the point p=t!r, is on the cone «S'.(/)— ot) (t)w(j)p = 0, 27. Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane. 28. Shew that the locus of the pole of the plane Sap = 1, with respect to the surface Sp(pp = 1, lf)4 QUATERNIONS. is a sphere^ if a be subject to the condition Sacl>-^a = C. 29. Shew that the equation of the surface generated by hnes drawn through the origin parallel to the normals to Sp^-'^p = 1 along its lines of intersection with Sp{<l> + F]--^P=zi, is m^ —^Sm {4, + ky^-m: = 0. 30. Common tangent planes are drawn to 2S\pSiJ,p + {p—Skij.)p^ = l, and Tp = k, find the value of A that the lines of contact with the former surface may be plane curves. What are they, in this case, on the sphere? Discuss the case of jo^—S^Xix = 0. 31. If tangent cones be drawn to Sp(j>^P = 1, from every point of 'Sp'pp = !> the envelop of their planes of contact is Sp^^p = 1. 32. Tangent cones are drawn from every point of S{p — a)(j>{p — a.) = n^, to the similar and similarly situated surface Sp4,p = 1, shew that their planes of contact envelop the surface {Sa(l)p-lY = n'^Sp(l)p. 33. Pind the envelop of planes which touch the parabolas p = ai^ + pt, p = aT^ + yr, where a, (3, y form a rectangular system, and t and t are scalars. 34. Find the equation of the surface on which lie the lines of contact of tangent cones drawn from a fixed point to a series of similar, similarly situatedj and concentric ellipsoids. 35. Discuss the surfaces whose equations are SapS^p = Syp, and S^ap + S.a^p— \. 36. Shew that the locus of the vertices of the right cones which touch an ellipsoid is a hyperbola. 37. If oj, Og, ag be vector conjugate diameters of Sp(f>p = 1, where ^^— %<^^ +%(^— m = 0, shew that 2(a^)=--^> 2(Fa, 0,)^= -} S^.(ua,a~= > ^ ' m V 1 Z^ ^ "12 3 ^ and 2 (<ii>af — m^ . CHAPTER IX. GEOMETRY OF CUEVES AND SURFACES. 279.] We have already seen (§31 (l)) that the equations p = ct>t = S.a/it), and p = (p{i, u) = 1.af{t, u), where a represents one of a set of given vectors, and /"a scalar func- tion of scalars t and u, represent respectively a curve and a surface. We commence the present too brief Chapter with a few of the im- mediate deductions from these forms of expression. We shall then give a number of examples, with little attempt at systematic devel- opment or even arrangement. 280.] What may be denoted by t and u in these equations is, of course, quite immaterial : but in the case of curves, considered geometrically, t is most conveniently taken as the length, s, of the curve, measured from some fixed point. In the Kinematical in- vestigations of the next Chapter t may, with great convenience, be employed to denote time. 281.] Thus we may write the equation of any curve in space as P = <i>^, where <^ is a vector function of the length, s, of the curve. Of course it is only a linem- function when, the equation (as in § 31 {I)) represents a straight line. 283.] We have also seen (§§ 38, 39) that ^P ^ A. A.' is a vector of unit length in the direction of the tangent at the ex- tremity of p. At the proximate point, denoted by "s + hs, this unit tangent vector becomes ^'s + (|)"s 6« + &e. 156 QUATERNIONS. [283. But., because T<})'s = 1, we have S.<j)'s (j/'s = 0. Hence ij/'s is a vector in the osculating plane of the curve, and per- pendicular to the tangent. Also, if bd be the angle bet.ween the successive tangents (j/s and <f/s + (p"s bs + , we have <^ = ^*'" so that t&e tensor of <^"s is the reciprocal of the radius of absolute curvature at the point s. 283.] Thus, if OP = (/>« be the vector of any point P of the curve, and if C be the centre of curvature at P, we have and thus OC = <hs jj- cf) s is the equation of the locus of the centre of curvature. Hence also F.cjj'skj/'s or <f>s^"s is the vector perpendicular to the osculating plane ; and T^^{4>'sU<^"s) is the tortuosity of the given curve, or the rate of rotation of its osculating plane per unit of length. 284.J As an example of the use of these expressions let us fin^ the curve whose curvature and tortuosity are both constant. We have curvature = T^"s = Tp"= c. Hence (j)'s(j/'s = p'p"= ca, where a is a unit vector perpendicular to the osculating plane. This o r fff . ffo ^ Pi Oct ^j ff ff pp +p ^=c^— = cc^Up =Cip , if Cj represent the tortuosity. Integrating we get p'p"- g^p'^^^ (1) where /3 is a constant vector. Squaring both sides of this equation, we get c2 = cf -/32 - 2 c^Sfip' (for by operating with S.p' upon (1) we get +c^ = Sj3p'), or Tj3 = ^/c^+cl 285.] GEOMETRY OF CURVES AND SURFACES. 157 Multiply (1) by p, remembering that Tp'= 1, and we obtain _ p" = _ q 4- p'^^ or, by integration, p = c-^s—pP-\-a, (2) where a is a constant quaternion. Eliminating p', we have of which the vector part is p"— p/32 = —cjsfi— Fafi. The complete integral of this equation is evidently P = ieos.sT^ + r,sin.sTl3-~{c^sl3+ Faj3), (3) f and T] being any two constant vectors. We have also by (2), Sfip = CjS + Sa, which requires that Sfi^ = 0, Sfirj = 0. The farther test, that Tp'=l, gives us -1 = Tl3\i^sin\sT^ + r,^cos\sT^-2Sir,sm.sTl3eos.sTl3)- -/^ • This requires^ of course^ so that (3) becomes the general equation of a helix traced on a right cylinder. (Compare § 31 (m).) 285.] The vector perpendicular from the origin on the tangent to the curve p = rf)« is, of course, -,Vp'p, or p'Fpp' (since p' is a unit vector). To find a common property of curves whose tangents are all equi- distant from the origin. Here TFpp'^z c, which may be written —p^—S^pp'=c^ (1) This equatiod shews that, as is otherwise evident, every curve on a sphere whose centre is the origin satisfies the condition. For ob- viously —p^ = c^ gives Spp'= 0, and these satisfy (1). If Spp' does not vanish, the integral of (1) is VTp^-c^ = s, (2) an arbitrary constant not being necessary, as we may measure s from any point of the curve. The equation of an involute which commences at this assumed point is -ST = p — sp'. 158 QUATERNIONS. [286. This gives T^^ = Tp^ + s^ + 2 sSpp' = Tp^^s''-2s^/Tp^-c^, by(l), = o\ by (2). This includes all curves whose involutes lie on a sphere about the origin. 286.] Find the locus of the foot of the ^perpendicular drawn to a tangent to a right helix from a point in the axis. The equation of the helix is p = acos- +/3sin- 4-y*, a a ' where the vectors a, ^, y are at right angles to each other, and Ta = Tl3=z h, while aTy = ^a^-h^. The equation of the required locus isj by last section, ■ar = p'Vpp' , s a^—l^ . *\ ^/ . s a'^—W' S\ b^ = a (cos — I 5 — ssin-) + fl(sm 5 — «cos-) + y-^-*. ^ a a^ a^ ^ a a^ a' ' a^ This curve lies on the hyperboloid whose equation is B'^aTn-^-S^^vs-a^S^yw = «*, as the reader may easily prove for himself. 287.] To find the least distance between consecutive tangents to a tortuous curve. Let one tangent be ct = p + xp' , then a consecutive one, at a distance hs along the curve, is ^ = p + p'6« + p"g +&c.+y(p' + /'85 + p"'g +...). The magnitude of the least distance between these lines is, by §§203,210, ^.(p'8* + p"g+p'"j^+...)C^r.p'(p' + p"6* + p"'g + ...) ~ Trp'p"is if we neglect terms of higher orders. It may be written, since p'p" is a vector, and Tp' = 1 , ^^.C/p'TpV". But (§133(2)) ^^^ = r^5s=p,p'S.p'py' Hence pj-,8.Up"rp'p"' 289.J GEOMETRY OP CURVES AND SURFACES. 159 is the small angle, 6</), betwee:rtlie two successive positions of the osculating plane. [See also § 283.] Thus the shortest distance between two consecutive tangents is expressed by the formula bcfibs^ 12/ ' vhere r, = -=y-, , is the radius of absolute curvature of the tortuous curve. 288.] Let us recur for a moment to the equation of the parabola (§31(/.)) ^ /3<2 P = "'^ + 2 ■ Here p'= {a + fit)-j-, whence, if we assume Safi = 0, from which the length of the are of the curve can be derived in terms of t by integration. Again, p"=(a+,0£+K|)^- dH _ d 1 _ dt S.^{a + pi) ^ ds^ ~ ds ' T{a +/3i!) ~ "•" S T{a + ^tf ' and therefore, for the vector of the centre of curvature we have (§ 283), ^^^f_^§^ -{a^ + ^H^f{-^o? + afiH)-\ which is the quaternion equation of the evolute. 289.] One of the simplest forms of the equation of a tortuous curve is fl^2 yp P = -i + '^ + \' where a, /3, y are any three non-coplanar vectors, and the numerical factors are introduced for convenience. This curve lies on a para- bolic cylinder whose generating lines are parallel to y ; and also on cylinders whose bases are a cubical and a semi-cubical parabola, their generating lines being parallel to ^ and a respectively. We have by the equation of the curve 160 QUATERNIONS. 29O. from which, by 2'/=!, the length of the curve can be fouud In terms of t ; and from which p" can be expressed in terms of s. The investigation of various properties of this curve is very easy, and will be of great use to the student. {Note. — It is to be observed that in this equation t cannot stand for *, the length of the curve. It is a good exercise for the student to shew that such an equation as or even the simpler form p- a^ + ^s^, involves an absurdity.] 290.] The equation p = <f)^€, where cf) is a. given self-conjugate linear and vector function, t a scalar variable, and e an arbitrary vector constant, belongs to a curious class of curves. We have at once — = ^' log (jbe, where \og<p is another self-conjugate linear and vector function, which we may denote by x- These functions are obviously commu- tative, as they have the same principal set of rectangular vectors, hence we may write gp which of course gives -j^ = x^Pt &c., since x does not involve t. As a verification, we should have ./,-"e = p+^a^ + ^— + &C. = (i + s^x+|Jx^+ )p where e is the base of Napier's Logarithms. This is obviously true if ^" = e*''', or (jb = gXj or log = X. which is our assumption. [The above process is, at first sight, rather startling, but the 293- GEOMETEY OF CURVES AND SUEFACES. 161 student may easily verify it by writing, in accordance with the , ilts of Chapter V, whence ^«e z= —g[aSaf—gl^^S^e—glySye. He will find at once X« = —logg^^aSat - hgg^pSfie-loggsySye, and the results just given follow immediately.] 291.] That the equation p = (^ (i5, w) = 2 . af{t, u) represents a surface is obvious from the fact that it becomes the equation of a definite curve whenever either t ov u has a particular value assigned to it. Hence the equation at once furnishes us with two systems of curves, lying wholly on the surface, and such that one of each system can, in general, be drawn through any assigned point on the surface. Tangents drawn to these curves at their point of intersection must^ of course, lie in the tangent plane, whose equation we have thus the means of forming. 292.] By the equation we have * = (§)*+(£)*' where the brackets are inserted to indicate partial differential coefii- cients. If we write this as dp = (ji'f df + (j)\ du, the normal to the tangent plane is evidently and the equation of that plane &(^-</))<^>'„=0. 293.] As a simple example, suppose a straight line to move along a fixed straight line, remaining always perpendicular to it, while rotating about it through an angle proportional to the space it has advanced ; the equation of the ruled surface described will evidently be p = at+u(PGOst + ysmt), fl) where a, j8, y are rectangular vectorSj and T0 = Ty. This surface evidently intersects the right cylinder p = a (p cos t + y sin t) + va, in a helix (§§ 31 (m), 284) whose equation is p = o^ + a(/3cos^ + ysini!). These equations illustrate very well the remarks made in §§ 3 1 (^, 29 1 M 162 QUATERNIONS. [294. as to the curves or surfaces represented by a vector equation ac- cording as it contains one or two scalar variables. From (1) we have dp = \a—u{^svo.t—y<iOst)'\dt-\-{^ cos t-\-y&mt)du, so that the normal at the extremity of p is Ta {y cost-p sin t) - uT^^ Ua. Hence, as we proceed along a generating line of the surface, for which t is constant, we see that the direction, of the normal changes. This, of coursCj proves that the surface is not developable. 294.] Hence the criterion for a developable surface is that if it be expressed by an equation of the form p = <f)t-\- tixj/t, where (j)t and \jft are vector functions, we must have the direction of the normal 'F{<t)'t + wft} \j/t independent of u. This requires either F-fi-^'t = 0, which would reduce the surface to a cylinder^ all the generating lines being parallel to each other ; or F(j>'t\}/t = 0. This is the criterion we seek, and it shews that we may write, for a developable surface in general, the equation p = (pt + U(p't • (1) Evidently p = ^t is a curve (generally tortuous) and (f/t is a tangent vector. Hence a developable surface is the locus of all tangent lines to a tortuous curve. Of course the tangent plane to the surface is the osculating plane at the corresponding point of the curve ; and this is indicated by the fact that the normal to (1) is parallel to r(i>tcl>"t. (See § 283.) To find the form of the section of the surface made iy a normal plane through a point in the curve. The equation of the surface is OT = p+«/3' + — p" + &c.+«(p'+*p'' + &c.). The part of tsr— p which is parallel to p' is -p'^(^-p)p'=-/(-(s+«^)-p"^(4+^) + ...); therefore ^-p = Ap'+(~+ws) p"-(~ +'^) p'FpY' + ... . 297-] GEOMETEY OP CUBVES AND SURFACES. 163 And, when A = 0, i.e. in the normal section, we have approximately so that ^ _ p = _ i_ p" _ 1_ p' fp'p'". Z o Hence the curve has an equation of the form a semicubical parabola. 395. J A Geodetic line is a curve drawn on a surface so that its osculating plane at any point contains the normal to the surface. Hence, if v be the normal at the extremity of p, p and p" the first and second differentials of the vector of the geodetic, S.vp'p"= 0, which may be easily transformed into V.vdUp'^ 0. 296.] In the sphere Tp = ayte, have V Up, hence S.pp'p"= 0, which shews of course that p is confined to a plane passing through the origin, the centre of the sphere. For a formal proof, we may proceed as follows — The above equation is equivalent to the three S9p = 0, Sdp'= 0, Sdp"= 0, from which we see at once that 5 is a constant vector, and therefore the first expression, which includes the others, is the complete in- tegral. Or we may proceed thus — = -pS.ppY+p"s.py= r. Vpp'rpp"= r. Vpp'dVpp', whence by § 133 (2) we have at once UVpp'= const. = suppose, which gives the same results as before. 297.] In any cone we have, of course, Svp = 0, since p lies in the tangent plane. But we have also Svp'= 0. Hence, by the general equation of § 295, eliminating v we get = S.pp'rp'p"= SpdUp' by § 133 (2). Integrating C=Sp Up'-jsdp Up'= Sp Up' +J Tdp. The interpretation of this is, that the length of any arc of the geo- detic is equal to the projection of the side of the cone (drawn to its 164 QTJATEENIONS. [298. extremity) upon the tangent to the g«odetic. In other words, when the cone is developed the geodetic becomes a straight line. A similar result may easily be obtained for the geodetic lines on any develop- able surface whatever. 298.] To find the shortest line connecting two points on a given surface. Here / Tdp is to be a minimum, subject to the condition that dp lies in the given surface. Now h^Tdp = fbTdp = -f^^^ = -fs. Udpdbp = - [_S. Udp 8/)] + fs.bpdUdp, where the term in brackets vanishes at the limits, as the extreme points are fixed, and therefore 8p = 0. Hence our only conditions are P ' S.bpdUdp = 0, and Svbp = 0, giving V.vdVdp = 0, as in § 295. If the extremities of the curve are not given, but are to lie on given curves, we must refer to the integrated portion of the ex- pression for the variation of the length of the arc. And its form S.Udpbp shews that the shortest line cuts each of the given curves at right angles. 299.] The osculating plane of the curve p^^t is S.4,'t<i,"t{m-p) = 0, (1) and is, of course, the tangent plane to the surface p = <t)t + U(t>'t (2) Let us attempt the converse of the process we have, so far, pursued, and endeavour to find (2) as the envelop of the variable plane (1). Differentiating (1) with respect to t only, we have By this equation, combined with (1), we have ^-p\\r.r<t>'rr<i>T\\<i^', or zT = p + u(l)'= (l)+'U(l/, which is equation (2). 300.] This leads us to the consideration of envelops generally, and the process just employed may easily be extended to the problem 302.J GEOMETRY OF CURVES AND SURFACES. 165 of finditiff the envelop of a series of surfaces whose equation contains one scalar parameter. When the given equation is a scalar one^ the process of finding the envelop is precisely the same as that employed in ordinary Cartesian geometry, though the work is often shorter and simpler. If the equation be given in the form p =-\}i{t, u, v), where t/^ is a vector function, t and u the scalar variables for any one surface, v the scalar parameter, we have for a proximate surface Pi = V' {h> %. ^i) = p+'Vt^t + 'Vu^'^'^Vv^'"- Hence at all points on the intersection of two successive surfaces of the series we have which is equivalent to the following scalar equation connecting the quantities t, u, and v ; This equation, along with p, = -f{i, u, v), enables us to eliminate t, u, v, and the resulting scalar equation is that of the required envelop. 301.] As an example, let us find the envelop of the osculating plane of a tortuous curve. Here the equation of the plane is (§ 299), S.{m-p)<i/t<i>"t= 0, or CT = (l>t+x^'t+i/^"t = •^{x,y, {), if p = <f)t be the equation of the curve. Our condition is, by last section, or S.<i>'t 4>"t l(t)'t + so4>"t + y ^'"t] = 0, or y84't<^"t<^"'t=(i. Now the second factor cannot vanish, unless the given curve be plane, so that we must have and the envelop is 'si =■ <pt + w<^'t the developable surface, of which the given curve is the edge of regression, as in § 299. 302.] When the equation contains two scalar parameters its differential coefiieients with respect to them must vanish, and we have thus three equations from which to eliminate two numerical quantities. 166 QUATERNIONS. [303. A very common form in whieli these two parameters appear ia quaternions is that of an unknown unit-vector. In this case the problem may be thus stated — Find the envelop of the surface whose scalar equation is Jpu^ a) = 0, wJiere a is subject to the one condition Ta = 1. Differentiating with respect to o alone, we have Svda = 0, Sada = 0, where v is a known vector function of p and a. Since da may have any of an infinite number of values, these equations shew that Fav = 0. This is equivalent to two scalar conditions only, and these, in addi- tion to the two given scalar equations, enable us to eliminate a. With the brief explanation we have given, and the examples which follow, the student will easily see how to deal with any other set of data he may meet with in a question of envelops. 303.] Find the envelop of a plane whose distance from the origin is constant. Here Sap =-—c, with the condition Ta = 1 . Hence, by last section, Vpa = 0, and therefore p = ca, or Tp = c, the sphere of radius c, as was to be expected. If we seek the envelop of those only of the planes which are parallel to a given vector /3, we have the additional relation Sa^ = 0. In this case the three differentiated equations are Spda = 0, Sada = 0, SjSda = 0, and they give S.a^p = 0. Hence a = U.^T^p, and the envelop is TVfip = cTfi, the circular cylinder of radius c and axis coinciding with fi. By putting Safi = e, where e is a constant different from zero, we pick out all the planes of the series which have a definite in- clination to j8, and of course get as their envelop a right cone. 304.] The equation S'^ap+tS.a^p = h represents a parabolic cylinder, whose generating lines are parallel to the vector aFa/S. For the equation is of the second degree, and 305.] GEOMETRY OP CURVES AND SURFACES. 167 • is not altered by increasing p by the vector xaFa^ ; also the surface cuts planes perpendicular to a in one line, and planes perpendicular to FajS in two parallel lines. Its form and position of course depend upon the values of a, /3, and 6. It is required to find its envelop if ^ and b be constant, and a be subject to the one scalar condition Ta=l. The process of § 302 gives, by inspection, pSap+ Vfip = oca. Operating by S.a, we get S^ap + S.aj3p =—«!, which gives S.a/Sp = x-i- i. But, by operating successively by S. Fj3p and by S.p, we have {FPpf = (vS.aISp, and {p^—x)Sap = 0. Omitting, for the present, the factor Sap, these three equations give, by elimination of x and a, {rppf = p^{p^+b), which is the equation of the envelop required. This is evidently a surface of revolution of the fourth order whose axis is /3 ; but, to get a clearer idea of its nature, put and the equation becomes {V^taf = c* + 6zt^, which is obviously a surface of revolution of the second degree, referred to its centre. Hence the required envelop is the reciprocal of such a surface, in the sense that t^e rectangle under the lengths of condirectional radii of the two is constant. We have a curious particular case if the constants are so related that b + ^^ =zQ, for then the envelop breaks up into the two equal spheres, touching each other at the origin, P^ = ± ^^Pi while the corresponding surface of the second order becomes the two parallel planes S^.^ = + e^. 305.] The particular solution above met with, viz. Sap — 0, limits the original problem, which now becomes one of finding the envelop of a line instead of a surface. In fact this equation, taken in conjunction with that of the parabolic cylinder, belongs to that generating line of the cylinder which is the locus of the vertices of the principal parabolic sections. 168 QUATERNIONS. [306. Our equations become 2S.al3p = h, Sap = 0, Ta = I; whence Ffip = ica, giving ^^ 2 and thence ^^fip = - ', so that the envelop is a circular cylinder whose axis is /3. [It is to be remarked that the equations above require that Sa^ = 0, so that the problem now solved is merely that of tke envelop of a parabolic cylinder which rotates about its focal line. This discussion has been entered into merely for the sake of explaining a peculiarity in a former result, because of course the present results can be obtained immediately by an exceedingly simple process.] 306.] The equation SapS.ajip = a^, with the condition Ta= I, represents a series of hyperbolic cylinders. It is required to find their envelop. As before, we have pS.app+ F/SpSap = xa, which by operating by S.a, S.p, and S. Vfip, gives 2a^ =—x, p^S.afip = xSap, {rfipySap=xS.afip. Eliminating a and x we have, as the equation of the envelop, p^iFjSpf = 4.a*. Comparing this with the equations p^=-2a^, and {rppY = -2a^, which represent a sphere and one of its circumscribing cylinders, we see that, if eondirectional radii of the three surfaces be drawn from the origin, that of the new surface is a geometric mean be- tween those of the two others. 307.] Find the envelop of all spheres which touch one given line and have their centres in another. Let p = ^-\-yy be the line touched by all the spheres, and let xa be the vector of the centre of any one of them, the equation is (by § 200, or § 201) y'^ip-xaf =-{r.y{fi-xa)Y, 3o8.] GEOMETEY OF CUEVES AND SURFACES. 169 • ov, putting for simplicityj but without loss of generality, Ty=l, Sa^ = 0, iSl3y = 0, so that /3 is the least vector distance between the given lines, {p—xa)^ = {^—xa)^-\-x'^S'^ay, and, finally, P^-fi^- ix Sap = x^ S^ay. Hence, by § 300, —2Sap = 2xS^ay. [This gives no definite envelop if Say = 0, i. e. if the line of centres is perpendicular to the line touched by all the spheres.] Eliminating x, we have for the equation of the envelop which denotes a surface of revolution of the second degree, whose axis is a. Since, from the form of the equation, Tp may have any magnitude not less than T^, and since the section by the plane Sap = is a real circle, on the sphere the surface is a hyperboloid of one sheet. [It will be instructive to the student to find the signs of the values of ^1,^2) ffs ^^ i^ § ^^^j ^^^ thence to prove the above con- clusion.] 308.] As a final example let us find the envelop of the hyperbolic cylinder SapS^p—o = 0, where the vectors a and /3 are subject to the conditions Ta = T^^ 1, Say = 0, aS^8 = 0, y and 6 being given vectors. [It will be easily seen that two of the six scalars involved in a, /3 still remain as variable parameters.] We have Sada = 0, Syda = 0, so that da = xVay. Similarly ^/3=yFj35. But, by the equation of the cylinders, SapSpd/S + SpdaSfip = 0, or ySapS.^hp +xS.aypSfip = 0. Now by the nature of the given equation, neither Sap nor S^p can vanish, so that the independence of da and d^ requires S.ayp = 0, S.fibp = 0. 170 QUATEE,]SriONS. [309- Hence a = U.y Fyp, fi =U.h Ftp, and the envelop is T.FypFbp — cTyb = 0, a surface of the fourth order^ which may be constructed by laying off mean proportionals between the lengths of condirectional radii of two equal right cylinders whose axes meet in the origin. 309.] "We may now easily see the truth of the following general statement. Suppose the given equation of the series of surfaces, whose envelop is required, to contain m vector, and n scalar, parameters ; and that the latter are subject top vector, and q scalar, conditions. In all there are 3m +n scalar parameters, subject to 3p + q scalar conditions. That there may be an envelop we must therefore in general have {3m + n) — {3_p + q) = 1, or = 2. In the former case the enveloping surface is given as the locus of a series of curves, in the latter of a series ot points. Differentiation of the equations gives us 3j) + q+l equations, linear and homogeneous in the 3m+n differentials of the scalar parameters, so that by the elimination of these we have one final scalar equation in the first case, two in the second ; and thus in each case we have just equations enough to eliminate all the arbitrary parameters. 310.] To find the locus of the foot of the perpendicular drawn from the origin to a tangent plane to any surface. If Svdp = be the differentiated equation of the surface, the equation of the tangent plane is S(T!r — p)v=0. We may introduce the condition Svp = 1, which in general alters the tensor of v, so that v~^ becomes the required vector perpendicular, as it satisfies the equation Smv = 1 . It remains that we eliminate p between the equation of the given surface, and the vector equation The result is the scalar equation (in vr) required. For example, if the given surface be the ellipsoid ^p4>P = 1. we have ■sr"^ = v = 4>p, 3 1 3-] GEOMETRY OF CUEVES AND SURFACES. 171 • so that the required equation is or /Sar^-V = OT*, which is Fresnel's Surface of Elasticity. (§ 263.) It is well to remark that this equation is derived from that of the reciprocal ellipsoid Sp(b-''-p = 1 by putting ot~^ for p. 3 11. J To find the reciprocal of a given surface with respect to the unit sphere whose centre is the origin. With the condition 8pv = 1, of last section, we see that — u is the vector of the pole of the tangent plane S{vT-p)v =(). Hence we must put zj=—v, and eliminate phj the help of the equation of the given sm-faee. Take the ellipsoid of last section, and we have so that the reciprocal surface is represented by It is obvious that the former ellipsoid can be reproduced from this by a second application of the process. And the property is general, for Spv = 1 gives, by differentiation, and attention to the condition Svdp = 0, the new relation Spdv = 0, so that p and r are corresponding vectors of the two surfaces : either being that of the pole of a tangent plane drawn at the extremity of the other. 312.] If the given surface be a cone with its vertex at the origin, we have a peculiar case. For here every tangent plane passes through the origin, and therefore the required locus is wbolly at an infinite distance. The difficulty consists in Spv becoming in this case a numerical multiple of the quantity which is equated to zero in the equation of the cone, so that of course we cannot put as above Spv = 1. 313.] The properties of the normal vector v enable us to write the partial differentia] equations of families of surfaces in a very simple- form. Thus the distinguishing property of Cylinders is that all their 172 QUATERNIONS. ' [SH- generating lines are parallel. Hence all positions of v must be parallel to a given plane — or Sav = 0, which is the quaternion form of the well-known equation ,dF dF dF „ dx dy dz To integrate it, remember that we have always Svdp = 0, and that as v is perpendicular to a it may be expressed in terms of any two vectors, /3 and y, each perpendicular to a. Hence v = x^ + yy, and xS^dp + ySydp = 0. This shews that S^p and Syp are together constant or together variable, so that SfSp =f{Syp), where/" is any scalar function whatever. 314.] In Surfaces of Bevolution the normal intersects the axis. Hence, taking the origin in the axis a, we have S.apv = 0, or V = xa + yp. Hence xSadp + ySpdp = 0, whence the integral Tp =f{Sap). The more common form, which is easily derived from that just written, is TFap = F{Sap). In Cones we have Svp = 0, and therefore Svdp = S.v{TpdUp+ UpdTp) = TpSvdUp. Hence SvdUp = 0, so that V must be a function of Up, and therefore the integral is AUp) = 0, which simply expresses the fact that the equation does not involve the tensor of p, i. e. that in Cartesian coordinates it is homogeneous. 315.] If equal lengths he laid off on the normals drawn to any surface, the new surface formed hy their extremities is normal to the same lines. For we have w = p + a Uv, and SvdTn = Svdp + aSvdUv = 0, which proves the proposition. Take, for example, the surface Sp(l>p = 1 ; 3 1 7-] GEOMETRY OP CURVES AND SURFACES. 173 the above equation becomes so that ^'=(^ + and the equation of the new surface is to be found by eliminating ~— (written ») between the equations 1 = <S'.(«(j!)+l)-i,!r<^(a;0+l)-iOT, -1 a and i=S4 (xcj) + 1 )-^zj(j) {xs^ + 1 y-'^-nr. 316.] It appears from last section that if one orthogonal surface can be drawn cutting a given system of straight lines, an inde- finitely great number may be drawn: and that the portions of these lines intercepted between any two selected surfaces of the series are all equal. Let p = a+XT, where o- and t are vector functions of p, and x is any scalar, be the general equation of a system of lines : we have Srdp = = S{p—a)dp as the differentiated equation of the series of orthogonal surfaces, if it exist. Hence the following problem. 317.] It is required to find the criterion of integrahility of the equation Svdp = (1) as the complete differential of the equation of a series of mrfaces. Hamilton has given [Elements, p. 702) an extremely elegant solu- tion of this problem, by means of the properties of linear and vector functions. We adopt a different and somewhat less rapid process, on account of some results it offers which will be useful to us in the next Chapter ; and also because it will shew the student the connection of our methods with those of ordinary differential equa- . tions. If we assume Fp= C to be the integral, and apply to it the very singular operator de- Adsed by Hamilton, „ . d . d , d dx '' dy dz ^ .dF .dF ,dF we have vi^= .^ +^^ +^^- 174 QUATERNIONS. [3 1 8. But p = ix +jy + hz, whence dp ■= idx+jdy-\-kdz, ,^ dF , dF , dF , „,„T, and Q = dF=-rdx-\--^dy-ir^rdz——SdpVF. dx dy ^ dz '^ Comparing with the given equation, we see that the latter repre- sents a series of surfaces if p, or a scalar multiple of it, can be ex- pressed as VF. If v = VF, „,-^ ^d^F d'^F d^Fs we have ^^ = V^^=-(^ + ^ + ^) ' a well-known and most important expression, to which we shall return in next Chapter. Meanwhile we need only remark that the last-written quantities are necessarily scalars, so that the only requisite condition of the integrability of (1) is rVv= (2) If V do not satisfy this criterion, it may when multiplied by a scalar. Hence the farther condition rv (wv) = 0, which may be written FvVw—wrVv = (3) This requires that SvVv = (4) If then (2) be not satisfied, we must try (4). If (4) be satisfied to will be found from (3) ; and in either case (1) is at once integrable. [If we put dv = (t>dp where </> is a linear and vector function, not necessarily self-con- jugate, we have rvv=:r(i^ + ...) = rii<t,i+...)=-e, by § 173. Thus, if (j) be self-conjugate, e = 0, and the criterion (2) is satisfied. If (j) be not self-conjugate we have by (4) for the cri- terion Sev= 0. These results accord with Hamilton's, lately referred to, but the mode of obtaining them is quite difierent from his.] 3I8.3 As a simple example let us first take lines diverging from, a point. Here v\[p, and we see that \i v = p Vz; = -3, so that (2) is satisfied. And the equation is Spdp = 0, whose integral Tp ■=■ C gives a series of concentric spheres. 3 1 9-] GEOMETEY OF CUEVES AND SUEPACBS. 175 Lines ^perpendicular to, and intersecting, it, fixed line. If a be the fixed line, ^ any of the others, we have S.a^p = 0, Sa^ =Q, Spdp = 0. Here i- \\ aVap, and therefore equal to it, because (2) is satisfied. Hence S.dpaVap = 0, or S.VapFadp = 0, whose integral is the equation of a series of right cylinders T^rap= C. 319.] To find the orthogonal trajectories of a series of circles whose centres are in, and their planes perpendicular to, a given line. Let a be a unit-vector in the direction of the line, then one of the circles has the equations Tp = G,\ Sap = C, 3 where G and C are any constant scalars whatever. Hence, for the required surfaces V II d^p II Fap, where d^p is an element of one of the circles, v the normal to the orthogonal surface. Now let dp be an element of a tangent to the orthogonal surface, and we have Svdp — S.apdp = 0. This shews that dp is in the same plane as a and p, i.e. that the orthogonal surfaces are planes passing through the common axis. [To integrate the equation S.apdp = evidently requires, by § 317^ the introduction of a factor. For rvFap = riirai+jVaj + Wak) = 2a, so that the first criterion is not satisfied. But S.FaprVFap = 2S.arap = 0, so that the second criterion holds. It gives, by (3) of § 317, F.Vu;Fap+2wa = 0, or pSaVto — aSpVw + 2 wa = 0. That is SaVw = 0, \ SpVw = 2w. J These equations are satisfied by But a simpler mode of integration is easily seen. Our equation may be written = S.aF^ = Sa^-fi = ^.^alog^Z-^ p Up p 176 QUATERNIONS. [320. which is immediately integrable, j3 being an arbitrary but constant vector. As we have not introduced into this work the logarithms of ver- sors, nor the corresponding angles of quaternions, we must refer to Hamilton's Elements for a farther development of this point.] 320.] To jmd the orthogonal trajectories of a given series of sur- faces. If the equation Fp = C, give Svdp — f^, the equation of the orthogonal curves is Vvdp = 0. This is equivalent to two scalar differential equations (§ 197), which, when the problem is possible, belong to surfaces on each of which the required lines lie. The finding of the requisite criterion we leave to the student. Let the surfaces be concentric spheres. Here p^ ^ g. and therefore Vpdp= 0. Hence Tp^ dUp=-Up Fpdp = 0, and the integral is Up = constant, denoting straight lines through the origin. Let the sv/rfaces be spheres touching each other at a common point. The equation is (§ 2 1 8) Sap-^ = G, whence V.papdp = 0. The integrals may be written S.aPp = 0, p^+hTVap = 0, the first (/3 being any vector) is a plane through the common dia- meter; the second represents a series of rings or tores (§323) formed by the revolution, about a, of circles touching that ILae at the point common to the spheres. Let the surfaces be similar, similarly situated, and concentric, sur- faces of the second order. Here Spxp = C, therefore ^XP^P = ^• But, by § 290, the integral of this equation is p = e'^e where (f> and x are related to each other, as in § 290 ; and e is any constant vector. 321.] GEOMETEY OF CURVES AND SUEPACES. 177 331 .J To integrate the linear partial differential equation of a family of surfaces. The equation (see § 3 1 3) dx dy dz ~ may be put in the very simple form S (o-V) V, ■= 0, if we write a- = iP+JQ + kB, 1 „ . d . d , d and V=t- — |-;---j.^--. dx '' dy dz This gives, at once, Vu = t/iFOcr, where »« is a scalar and 6 a vector (in whose tensor m might have been included, but is kept separate for a special purpose). Hence dit = — S{dpV)ii = —mS.dddp = —S.edr, if we put dT = mr.<Tdp so that m is an integrating factor of V. (rdp. If a value of m can be found, it is obvious, from the form of the above equation, that d must be a function of r alone ; and the integral is therefore w = F{t) = const, where F is an arbitrary scalar function. Thus the differential equation of Cylinders is ' S(dV)u=0, where a is a constant vector. Here m=l, and M = F{Fap) = const. That of Cones referred to the vertex is S{pV)u= 0. Here the expression to be made integrable is r.pdp. But Hamilton long ago shewed that (§133 (2)) dUp _ ydp _ V.pdp -W~ P~ {Tpf ' which indicates the value of m, and gives u = F{TJp) — const. It is obvious that the above is only one of a great number of different processes which may be applied to integrate the differential equation. It is quite easy, for instance, to pass from it to the assumption of a vector integrating factor instead of the scalar m, N 178 QUATERNIONS. [322. aud to derive tlie usual criterion of integrability. There is no diffi- culty in modifying the process to suit the case when the right-hand member is a multiple of u. In fact it seems to throw a very clear light upon the whole subject of the integration of partial differ- ential equations. If, instead of S (o-V), we employ other operators as S {(tV) S {tV), S.o-VtV, &c. (where V may or may not operate on u alone), we can pass to linear partial differential equations of the second and higher orders. Similar theorems can be obtained from vector operations, as V{<tV)*. 322.] Find the general equation of surfaces described by a line which always meets, at right angles, a fixed line. If a be the fixed line, y3 and y forming with it a rectangular unit system, then p = a;a +y + zy), where y may have all values, but x and z are mutually dependent, is one form of the equation. Another, expressing the arbitrary relation between x and z is But we may also write p = aF{x) +ya''P, as it obviously expresses the same conditions. The simplest case is when F{x) = hx. The surface is one which cuts, in a right helix, every cylinder which has a for its axis. 323.] The centre of a sphere moves in a given circle, find the equa- tion of the ring described. Let a be the unit-vector axis of the circle, its centre the origin, r its radius, a that of the sphere. Then [p-^f =-0^ is the equation of the sphere in any position, where <So/3 = 0, 2)3 = n These give S.a^p = 0, and ^ must now be eliminated. The result is that ^ = raUVap, giving (p^— r^-t-a^)^ = ^r^T'^Vap, = 4r^-p^-S^ap), which is the required equation. It may easily be changed to (p^-a'^ + r^)^ =-4:a^p^-4:rWap, ...: (1) and in this form it enables us to give an immediate proof of the very singular property of the ring (or tore) discovered by Villarceau. * Tait, Proc. R. S. E., 1869-70. 324.J GEOMETRY OF CURVES AND SURFACES. 179 For the planes S.p (a± ) = 0, which together are represented by r^{r^-a^)8^ap-a'^S^^p = 0, evidently pass through the origin and touch (and cut) the ring. The latter equation may be written r'^S^ap-a^{8^ap + S^pU^) = 0, or r^S^ap + a^{p^ + S^.apU^) =0 (2) The plane intersections of (1) and (2) lie obviously on the new surface (^2_^2 + y2)2 ^ ia^S^.apUia, which consists of two spheres of radius r, as we see by writing its separate factors in the form (p±aaUpf+r^ = 0. 334.] It may be instructive to work out this problem from a different point of view, especially as it affords excellent practice in transformations. A circle revolves about an axis passing within it, the perpendicular from the centre on the axis lying in the plane of the circle: shew that, for a certain position of the axis, the same solid mny he traced out by a circle revolving about an external axis in its own plane. Let a = •fh'^ + c^ be the radius of the circle, i the vector axis of rotation, —ca (where Ta=-\) the vector perpendicular from the centre on the axis i, and let the vector hi + da be perpendicular to the plane of the circle. The equations of the circle are (p_ca)2 + ^2 + c2 = 0, \ S(i + Yia)p = 0. C Also —p^ = S^ip + S^ap + S'^.iap, b^ = SHp + S^ap+ -^SHp by the second of the equations of the circle. But, by the first, (/)2 + 5Z)2 = 4c2/SV = -4 {c^p'^+a^SHp), which is easily transformed into {(?-¥f=-i.a^{p^ + S^ip), or p2_52 ^ —2aTrip. If we put this in the forms p^-h^ = 2aSpp, and {p-a^f + c^=:0, N a 180 QUATERNIONS. [32 5- where ;3 is a unit-vector perpendicular to i and in tlie plane of i and p, we see at once that the surface will be traced out by a circle of radius c, revolving about i, an axis in its own plane^ distant a from its centre. This problem is not well adapted to shew the gain in brevity and distinctness which generally follows the use of quaternions ; as, from its very nature, it hints at the adoption of rectangular axes and scalar equations for its treatment, so that the solution we have given is but little different from an ordinary Cartesian one. 325.] A surface is generated hy a straight line which intersects two fixed lines : find the general equation. If the given lines intersect, there is no surface but the plane con- taining them. Let then their equations be, p = a + xfi, p = a^ + XiPi- Hence every point of the surface satisfies the condition, § 30, p=y(a + a;^) + (l-5^)(ai + 3'i^i) (1) Obviously y may have any value whatever : so that to specify a particular surface we must have a relation between x and x^. By the help of this, x^ may be eliminated from (1), which then takes the usual form of the equation of a surface P = 't>i'«,^)- Or we may operate on (1) by F.(a + xj3-- ai—XiJ3i), so that we get a vector equation equivalent to two scalar equations (§§ 98, 116), and not containing y. From this x and x^ may easily be found in terms of p, and the general equation of the possible surfaces may be written /"{^t *i) = 0, where /" is an arbitrary scalar function, and the values of x and x^ are expressed in terms of p. This process is obviously applicable if we have, instead of two straight lines, any two given curves through which the line must ' pass ; and even when the tracing line is itself a given curve, situated in a given manner. But an example or two will make the whole process clear. 326.] Suppose the moveable line to le restricted by the condition that it is always parallel to a fixed plane. Then, in addition to (1), we have the condition Sy{a-i^-\-x-yP-^—a — x^) = 0, y being a vector perpendicular to the fixed plane. We lose no generality by assuming o and Oj, which are any 327.] GKOMETEY OF CURVES AND SUEFACES. 181 vectors drawn from the origin to the fixed lines, to be each per- pendicular to y ; for, if for instance we could not assume Sya = 0, it would follow that Sy^ = 0, and the required surface would either be impossible, or would be a plane, cases which we need not con- sider. Hence x^8y^^-x8y^ - 0. Eliminating' ajj, by the help of this equation, from (1) of last section, we have , „; , , ^ Sy& -. Operating by any three non-coplanar vectors and with the charac- teristic S, we obtain three equations from which to eliminate a; and y. Operating by S.y we find Syp = xSjSy. Eliminating x by means of this, we have finally ^■'(« + ^^)(«.+ ^) = «. which appears to be of the third order. It is really, however, only of the second order, since, in consequence of our assumptions, we have Vauj^ \\ y, and therefore Syp is a spurious factor of the left-hand side. 327.] Let the fixed lines he perpendicular to each other, and let the moveable line pass through the circumference of a circle, whose centre is in the common perpendicular, and whose plane bisects that line at right angles. Here the equations of the fixed lines may be written p = a + x^, p =— a+a?iy, where a, j3, y, form a rectangular system, and we may assume the two latter to be unit-vectors. The circle has the equations p^ =—a^, Sap = 0. Equation (1) of § 325 becomes p = i/{a+xj3} + {l-if){-a + x^y). Hence Sar'^p = y—(l—^] = 0, or y = i- Also p2= -«2 = (2y-l)2 a'-x^f-xl (1-^)^ or 4fl^ = (x^+xl), so that if we now suppose the tensors of /3 and y to be each 2 a, we may put x = cos 0, x^ = sin 6, from which p = (2j^— l)a + y/3cos0+(l— y)ysin5; ^•^^ ^^""^ {l+Sa-^pf + {l-Sa-^pf = '^ • 182 QUATBRNIONS. [328. For this very simple case the solution is not better than the ordinary Cartesian one; but the student will easily see that we may by very slight changes adapt the above to data far less sym- metrical than those from which we started. Suppose, for instance, /3 and y not to be at right angles to one another ; and suppose the plane of the circle not to be parallel to their plane, &c., &c. But farther, operate on every line in space by the linear and vector function (^, and we distort the circle into an ellipse, the straight lines remaining straight. If we choose a form of ^ whose principal axes are parallel to a, p, y, the data will remain symmetrical, but not unless. This subject will be considered again in- the next Chapter. 328.] To find the curvature of a normal section of a central surface of the second order. In this, and the few similar investigations which follow, it will be simpler to employ infinitesimals than differentials ; though for a thorough treatment of the subject the latter method, as may be seen in Hamilton's Elements, is preferable. We have, of course, '^/'</>P = Ij and, if p + hp be also a vector of the surface, we have rigorously, whatever be the tensor ofbp, Sip + 8p)<t>{p + bp)= 1. Hence 2Sbpcl)p-\-Sbp<j)bp = (1) Now </)p is normal to the tangent plane at the extremity of p, so that if t denote the distance of the point p + bp from that plane i =-SbpU(l)p, and (1) may therefore be written •itT<i>p-T^bpS.mp^Ubp = 0. But the curvature of thfe section is evidently "^ T^bp ' or, by the last equation, ±-^^s.mp<i>mp. In the limit, bp is a vector in the tangent plane ; let ct- be the vector semidiameter of the surface which is parallel to it, and the equation of the surface gives T^isS .U-stcjjU-st = 1, so that the curvature of the normal section, at the point p, in the direction of or, is 1 329-J GEOMETRY OP CURVES AND SURFACES. 183 • Hirectly as the perpendicular from the centre on the tangent plane, and inversely as the square of the semidiameter parallel to the tangent line, a well-known theorem. 329.] By the help of the known properties of the central section parallel to the tangent plane, this theorem gives us all the ordinary properties of the directions of maximum and minimum curvature, their being at right angles to each other, the curvature in any normal section in terms of the chief curvatures and the inclination to their planes, &c., &c., without farther analysis. And when, in a future section, we shew how to find an osculating surface of the second order at any point of a given surface, the same properties will be at once established for surfaces in general. Meanwhile we may prove another curious property of the surfaces of the second order, which similar reasoning extends to all surfaces. The equation of the normal at the point p + 8p in the surface treated in last section is CT- = /3 + 8p+«(^(p + 8/)) (1) This intersects the normal at p if (§§ 203, 210) S.hp^p^hp = 0, that is, by the result of § 273, if 8p be parallel to the maximum or minimum diameter of the central section parallel to the tangent plane. Let o-j and o-g be those diameters, then we may write in general hp =piTi + q(T2, where ^ and q are scalars, infinitely small. If we draw through a point P in the normal at p a line parallel to (Tj, we may write its equation OT = p-{-a(j)p+^a^. The proximate normal (1) passes this line at a distance (see § 203) S . {a(l>p — bp) UF(Ti (t){p + 8/)), or, neglecting terms of the second order, ,,,-p- ■ (op 84pu-i(i)iT-^ + aqS.(l)p<jj(p<T2 + q S.cria^fjyp). IT (r-j(pp The first term in the bracket vanishes because o-j is a principal vector of the section parallel to the tangent plane, and thus the expression becomes / a „ \ Hence, if we take a — Tel, ^^ distance of the normal from the new line is of the second order only. This makes the distance of P from the point of contact T(f>pT(Tl, i.e. the principal radius of curvature 184 QUATERNIONS. [330. along the tangent line parallel to o-g. That is, the group of normals drawn near a point of a central surface of the second order pass ulti- mately through two lines each parallel to the tangent to one principal section, and passing through the centre of curvature of the other. The student may form a notion of the nature of this proposition by con- sidering a small square plate, with normals dravra at every point, to he slightly bent, but by different amounts, in planes perpendicular to its edges. The first bending will make all the normals pass through the axis of the cylinder of which the plate now forms part ; the second bending will not sensibly disturb this arrangement, except by lengthening or shortening the line in which the normals meet, but it will make them meet also in the axis of the new cylinder, at right angles to the first. A small pencil of light, with its focal lines, presents this appearance, due to the fact that a series of rays originally normal to a surface remain normals to a surface after any number of reflections and refractions. (See § 315). 330.] To extend these theorems to surfaces in general, it is only necessary, as Hamilton has shewn, to prove that if we write dv = (\)dp, is a self-conjugate function ; and then the properties of <|), as ex- plained in preceding Chapters, are applicable to the question. As the reader will easily see^ this is merely another form of the investigation contained in § 317. But it is given here to shew what a number of very simple demonstrations may be given of almost all quaternion theorems. The vector v is defined by an equation of the form dfp = Svdp, where /" is a scalar function. Operating on this by another inde- pendent symbol of differentiation, 8, we have hdfp = Sbvdp + Svhdp. In the same way we have dbfp = Sdvhp + Svdbp. But, as d and 8 are independent, the left-hand members of these equations, as well as the second terms on the right (if these exist at all), are equal, so that we have Sdvbp = Shvdp. This becomes, putting dv = <^dp, and therefore Sv = ^6p, 8bp<pdp = Sdptjibp, which proves the proposition. 333-] GEOMETRY OP CURVES AND SURFACES. 185 331.] If we write the differential of the equation of a surface in the form df(t = iSvAp, then it is easy to see that f{p-\-dp) =fp+2Svdp + Sdvdp + kc., the remaining terms containing as factors the third and higher powers of Tdp. To the second order, then, we may write, except for certain singular points, = 2Svdp + Sdvdp, and, as before, (§ 328), the curvature of the normal section whose tangent line is dp is 1 „ dv Yv Tp' 333.] The step taken in last section, although a very simple one, virtually implies that the first three terms of the expansion of /(p + dp) are to be formed in accordance with Taylor's Theorem, whose applicability to the expansion of scalar functions of quater- nions has not been proved in this work, (see § 135); we therefore give another investigation of the curvature of a normal section, employing for that purpose the formulae of § (282). We have, treating dp as an element of a curve, Svdp = 0, or, making s the independent variable, Svp'= 0. From this, by a second dififerentiation, 8^p' + Svp"= 0. The curvature is, therefore, since v \\ p" and Tp'— \, 333.] Since we have shewn that dv ^ (f)dp where is a self-conjugate linear and vector function, whose con- stants depend only upon the nature of the surface, and the position of the point of contact of the tangent plane ; so long as we do not alter these we must consider if) as possessing the properties explained in Chapter V. Hence, as the expression for Tp" does not involve the tensor of dp, we may put for dp any unit-vector r, subject of course to the condition Svt = 0. , (1) And the curvature of the normal section whose tangent is r is 186 QUATERNIONS. [334- If we consider the central section of the surface of the second order &ss^^-\-Tv = 0, made by the plane Svm = 0, we see at once that the curvature of the given surface along the normal section touched hy t is inversely as the square of the parallel radius in the auxiliary surface. This, of course, includes Euler's and other well-known Theorems. 334.J To find the directions of maximum and minimum curvature, we have St<^t = max. or min. with the conditions^ Svt = 0, Tt= 1. By differentiationj as in § 273, we obtain the farther equation S.VT(\)T = (1) If T be one of the two required directions, t'=tUv is the other, for the last-written equation may be put in the form S.TUv(t>{vTUv) = 0, i.e. S.T'<t>{vT') = 0, or 8.v/^T = 0. Hence the sections of greatest and least curvature are perpendicular to one another. We easily obtain, as in § 273, the following equation S.v{(f)-\-ST^T)-'^V = 0, whose roots divided by Tv are the required curvatures. 335.] Before leaving this very brief introduction to a subject, an exhaustive treatment of which will be found in Hamilton's Elements, we may make a remark on equation (1) of last section S.VT(i)T = 0, or, as it may be written, by returning to the no'tation of § 333, S.vdpdv = 0. This is the general equation of lines of curvature. For, if we define a line of curvature on any surface as a line such that normals drawn at contiguous points in it intersect, then, bp being an element of such a line, the normals ■ST = p + xv and ■or = p + 5p + y (v + bv) must intersect. This gives, by § 203, the condition , S.bpvbv = 0, as above. EXAMPLES TO CHAPTER IX. 187 EXAMPLES TO CHAPTER IX. 1 . Find the length of any arc of a curve drawn on a sphere so as to make a constant angle with a fixed diameter. 2. Shew that, if the normal plane of a curve always contains a fixed line, the curve is a circle. 3. Mnd the radius of spherica,l curvature of the curve p = (jit. Also find the equation of the locus of the centre of spherical curvature. 4. (Hamilton, Bishop Law^ s Premium Examination, 1854.) (a.) If p be the variable vector of a curve in space, and if the differential Ak be treated as = 0, then the equation dT{p-K) = obliges K to be the vector of some point in the normal plane to the curve. (b.) In like manner the system of two equations, where dK and d^K are each = 0, dT(p-K) = 0, d^T{p-K) = 0, represents the axis of the element, or the right line drawn through the centre of the osculating circle, per- pendicular to the osculating plane. (c.) The system of the three equations, in which k is treated as constant, dT{p-K) = 0, d^T(p-K) = 0, d^T{p-K) = 0, determines the vector k of the centre of the osculating sphere. {d.) For the three last equations we may substitute the follow- ing : S.{p—K)dp = 0, S.{p-K)d\ + dp^ = 0, S.{p-K)d^p + 3S.dpd^p = 0. (e.) Hence, generally, whatever the independent and scalar variable may be, on which the variable vector p of the curve depends, the vector k of the centre of the oscu- lating sphere admits of being thus expressed : 3 F.dpd^pS.dpd^p-dp^ F.dpd^p K = p + S.dpd^pd^p 188 QUATEEKIONS. (/".) In general, d{d-W.dpUp) = d{Tp-^r.pdp) = Tp-'^ (sr.pdpS.pdp-p^r.pd^p) ; whence, ^r.pdpS.pdp-pW.pd^P = p^Tpd{p-^F.dpUp); and, therefore, the recent expression for k admits of being thus transformed, dp*d(dp-^r.d^pUdp ) "-P'^ S.d^pd^pUdp iff.) If the length of the element of the curve be constant, dTdp=0, this last expression for the vector of the centre of the osculating sphere to a curve of double curva- ture becomes, more simply^ d.d^pdp^ K = p + or K = p + S.dpd^pd^p ' F.d^pdp^ S.dpd^pd^p {h.) Verify that this expression gives /c = 0, for a curve de- scribed on a sphere which has its centre at the origin of vectors ; or shew that whenever dTp = 0, d^Tp = 0, d^Tp = 0, as well as dTdp = 0, then pS.dp-''d''pd^p=r.dpdy. 5. Find the curve from every point of which three given spheres ajjpear of equal magnitude. 6. Shew that the locus of a point, the difference of whose dis- tances from each two of three given points is constant, is a plane curve. 7. Find the equation of the curve which cuts at a given angle all the sides of a cone of the second order. Find the length of any are of this curve in terms of the distances of its extremities from the vertex. 8. Why is the centre of spherical curvature, of a curve described on a sphere, not necessarily the centre of the sphere ? 9. Find the equation of the developable surface whose generating lines are the intersections of successive normal planes to a given tortuous curve. 1 0. Find the length of an arc of a tortuous curve whose normal planes are equidistant from the origin. 11. The reciprocals of the perpendiculars from the origin on the tangent planes to a developable surface are vectors of a tortuous EXAMPLES TO CHAPTER IX. 189 curve ; from whose osculatin^planes the cusp-edge of the original surface may be reproduced by the same process. 12. The equation p=Fa'p, where a is a unit- vector not perpendicular to ft represents an ellipse. If we put y = Fa^, shew that the equations of the locus of the centre of curvature are S.pyp = 0, Sipp + S^yp = {fiSUapf. 13. Find the radius of absolute curvature of a spherical conic. 14. If a cone be cut in a circle by a plane perpendicular to a side, the axis of the right cone which osculates it, along that side, passes through the centre of the section. 15. Shew how to find the vector of an umbilicus. Apply your method to the surfaces whose equations are Spipp = 1, and SapS^pSyp = 1. 16. Find the locus of the umbilici of the surfaces represented by the equation Sp {(p + A)-^p=l, where A is an arbitrary parameter. 17. Shew how to find the equation of a tangent plane which touches a surface along a line^ straight or curved. Find such planes for the following- surfaces Spipp = 1, Sp{<j>-p^)-^p=l, and {p^-a'^ + b^y + 4:{a^p^ + 6^S'^ap)= 0. 18. Find the condition that the equation S{p + a)<l>p= 1, where ^ is a self-conjugate linear and vector function, may represent a cone. 19. Shew from the general equation that cones and cylinders are the only developable surfaces of the second order. 20. Find, the equation of the envelop of planes drawn at each point of an ellipsoid perpendicular to the radius vector from the centre. 21. Find the equation of the envelop of spheres whose centres lie on a given sphere, and which pass through a given point. 22. Find the locus of the foot of the perpendicular from the centre to the tangent plane of a hyperboloid of one, or of two, sheets. 190 QUATEENIONS. 23. 'H.arailtou, Mskqp Law's Premium Hxamination, 1852, {a.) If p be the vector of a curve in space, the 'length of the element of that curve is Tdp ; and the variation of the length of a finite arc of the curve is b/Tdp = -fSUdpbdp =-ASUdpbp+/SdUdpbp. (5.) Hence, if the curve be a shortest line on a given surface, for which the normal vector is v, so that Svbp = 0, this shortest or geodetic curve must satisfy the differential equation, FvdUdp = 0. Also, for the extremities of the arc, we have the limiting equations, SUdpo Spo = J SUdp^ 8pi = 0. Interpret these results, (c.) For a spheric surface, Fvp = 0, pdUdp=Q ; the integrated equation of the geodetics is p Udp = ■nr, giving Sxsp = (great circle). For an arbitrary cylindric surface, Sav = 0, adUdp = ; the integral shews that the geodetic is generally a helix, making a constant angle with the generating lines of the cylinder. [d.) For an arbitrary conic surface, Svp = 0, SpdUdp = ; integrate this differential equation, so as to deduce from it, TVpUdp = const. Interpret this result ; shew that the perpendicular from the vertex of the cone on the tangent to a given geo- detic line is constant ; this gives the rectilinear develop- ment. When the cone is of the second degree, the same property is a particular case of a theorem respecting confocal surfaces, (e.) For a surface of revolution, S.apv — 0, S.apdUdp = ; integration gives, const. = S.apUdp = TVapSU (Fap.dp) ; the perpendicular distance of a point on a geodetic line from the axis of revolution varies inversely as the cosine of the angle under which the geodetic crosses a parallel (or circle) on the surface. EXAMPLES TO CHAPTEE IX. 191 (/'.) The diiferential eqrration, S.apdUdp = 0, is satisfied not only by the geodeties, but also by the circles, on a surface of revolution ; give the explanation of this fact of calculationj and shew that it arises from the coinci- dence between the normal plane to the circle and the plane of the meridian of the surface. (g.) For any arbitrary surface, the equation of the geodetic may be thus transformed, S.vdpcPp = ; deduce this form, and shew that it expresses the normal property of the osculating plane. (A.) If the element of the geodetic be constant, dTdp = 0, then the general equation formerly assigned may be reduced to r.vd^p= 0. Under the same condition, d^p = —v'^Sdvdp. {i.) If the equation of a central surface of the second order be put under the form fp = I, where the function _/ is scalar, and homogeneous of the second dimension, then the diiferential of that function is of the forni dfp = 2S.vdp, where the normal vector, v = <l>p, is a dis- tributive function of p (homogeneous of the first dimen- sion), dv=d(j)p = <l)dp. This normal vector v may be called the vector of proximity (namely, of the element of the surface to the centre) ; because its reciprocal, v~^, represents in length and in direction the perpendicular let fall from the centre on the tangent plane to the surface. (^.) If we make S<T<^p =y(o-, p), this function/" is commutative with respect to the tvjo vectors on which it depends, f{p, a) =/'(*, p) ; it is also connected with the former functiony, of a single vector p, by the relation,/" (p, p) ■=fp : so that fp = Sp<pp. fdp = Sdpdv ; dfdp = 2S.dv d^p ; for a geodetic, with, con- stant element, 2jdp V this equation is immediately integrable, and gives const. =Tv-J{fJJdp) = reciprocal of Joachimstal's pro- duct, PB. (l.) If we give the name of " Didonia" to the curve (discussed by Delaunay) which, on a given surface and with a given perimeter, contains the greatest area, then for 192 QUATERNIONS. such a Didonian curve we have by quaternions the formula, fS. Uvdpbp + c h/Tdp = 0, where c is an arbitrary constant. Derive hence the differential equation of the second order, equivalent (through the constant c) to one of the third order, g-^Sp = F. UvdUdp. Geodeties are, therefore, that limiting case of Didonias for which the constant c is infinite. On a plane, the Didonia is a circle, of which the equation, obtained by integration from the general form, is p = ■uT + cUvdp, m being vector of centre, and c being radius of circle. (m.) Operating by 8. TJdp, the general differential equation of the Didonia takes easily the following forms : c'-'Tdp =S{Uvdp.dUdp); c-^Tdp^ = S{Uvdp.d^p); c-'^Tdp^ = S.Uvdpd^p; Uvdp {n.) The vector w, of the centre of the osculating circle to a curve in space, of which the element Tdp is constant, has for expression, dp'' Hence for the general Didonia, c"i = 5i Uvdp T{p-<^) = cSU'" vdp (o.) Hence, the radius of curvature of any one Didonia varies, in general, proportionally to the cosine of the inclination of the osculating plane of the curve to the tangent plane of the surface. And hence, by Meusnier's theorem, the difference of the squares of the curvatures of curve and surface is con- stant J the curvature of the surface meaning here the reciprocal of the radius of the sphere which osculates in the reduction of the element of the Didonia. {p.) In general, for any curve on any surface, if £ denote the vector of the intersection of the axis of the element (or EXAMPLES TO CHAPTER IX. 193 the axis of the circle osculating to the curve) with the tangent plane to the surface, then Hence, for the general Didonia, with the same significa- tion of the symbols, £ = p — cTIvdp ; and the constant c expresses the length of the interval p— f, intercepted on the tangent plane, between the point of the curve and the axis of the osculating circle. {q.) If, then, a sphere be described, which shall have its centre on the tangent plane, and shall contain the osculating circle, the radius of this sphere shall always be equal to c. [r.) The recent expression for ^, combined with the first form of the general differential equation of the Didonia, gives di = -crdUv Udp ; Vvd^ = 0. («.) Hence, or from the geometrical signification of the con- stant c, the known property may be proved, that if a developable surface be circumscribed about the arbitrary surface, so as to touch it along a Didonia, and if this developable be then unfolded into a plane, the curve will at the same time be flattened (generally) into a circular arc, with radius = c. 24. Find the condition that the equation Sp(<t>+f)-^P=l may give three real values of y for any given value of p. Ifybe a function of a scalar, parameter ^, shew how to find the form of this function in order that we may have ^ ^ dx^ ^ df ^-dz^ Prove that the following is the relation between / and ^, ,.=./• ^f =f^ ^ ^{9i+f)i9^+f){9z+f) ^ ^^f in the notation of § 147. 25. Shew, after Hamilton, that the proof of Dupin's theorem, that "each naember of one of three series of orthogonal surfaces cut? each member of each of the other series along its lines of curvature," may be expressed in quaternion notation as follows : 194 QUATERNIONS. If Svdp = 0, Sv'dp — 0, S.vv'dp = be integrable, and if Svv'= 0, then Fv'dp = 0, makes S.vv'dv = 0. Or, as follows, If SvVv = Q, Sv'Vv'=0, Sv"Vv"=:0, and r.w'v"= 0, then S.v"{Sv'V.v)=:0, 1 „ . d . d J d where V = i-r-+;T-+«-i-- dx dy dz 26. Shew that the equation Vap = pVfip represents the line of intersection of a cylinder and cone, of the second order, which have /3 as a common generating line. 27. Two spheres are described, with centres at A, B, where OA = a, OB — y3, and radii a, h. Any line, OFQ,, drawn from the origin, cuts them in T, Q respectively. Shew that the equation of the locus of intersection of AT, BQ has the form r{a + aU{p~a)) {fi + bU(p-fi)) = 0. Shew that this involves S.a^p = 0, and therefore that the left side is a scalar multiple of V.afi, so that the locus is a plane curve. Also shew that in the particular case Fal3 = 0, the locus is the surface formed by the revolution of a Cartesian oval about its axis. CHAPTER X. KINEMATICS. 336.] When a point's vector, p, is a function of the time t, we have seen (§36) that its vector- velocity is expressed by -j- or, in Newton's notation, by p. That is, if p = cpt be the equation of an orbit, containing (as the reader may see) not merely the form of the orbit, but the law of its description also, then p = ^'t gives at once the form of the Hodograph and the law of its de- scription. This shews immediately that the vector-cjcceleration of a point's motion, d^p df-"''' is the vector-velocity in the hodograph. Thus the fundamental pro- perties of the hodograph are proved almost intuitively. 337.] Changing the independent variable, we have dp ds , P^TsTt^''^' if we employ the dash, as before, to denote -5- • This merely shews, in another form, that p expresses the velocity in magnitude and direction. But a second differentiation gives p = vp' + v^p". This shews that the vector-acceleration can be resolved into two components, the first, vp', being in the direction of motion and equal in magnitude to the acceleration of the velocity, t; or -=- ; U/t the second, v^p", being in the direction of tha radius of absolute a 196 QUATERNIONS. [338. curvature, and having for its amount the square of the velocity multiplied by the curvature. [It is scarcely conceivable that this important fundamental pro- position, of which no simple analytical proof seems to have been obtained by Cartesian methods, can be proved more elegantly than by the process just given.] 338.] If the motion be in a plane curve, we may write the equation as follows, so as to introduce the usual polar coordinates, r and 6, zf p = ra"^, where a is a unit-vector perpendicular to, ^ a unit-vector in, the plane of the curve. Here, of course, r and may be considered as connected by one scalar equation ; or better, each may be looked on as a function of i. By differentiation we get 29 29 p = ra^'^ + rdaa'^ ^, which shews at once that r is the velocity along, rd that perpen- dicular to, the radius vector. Again, 2£ 29 which gives, by inspection, the components of acceleration along, and perpendicular to, the radius vector. 339.] For uniform acceleration in a constant direction, we have at once, • p = a. Whence p = ai + l3, where ^ is the vector-velocity at epoch. This shews that the hodograph is a straight line described uniformly. Also p = —-+fit, no constant being added if the origin be assumed to be the position of the moving point at epoch. Since the resolved parts of p, parallel to /3 and a, vary respect- ively as the first and second powers of i, the curve is evidently a parabola (§31 (/)). But we may easily deduce from the equation the following result, T(p + iPa-^^) =-SUa(p + ^ a-^) , the equation of a paraboloid of revolution, whose axis is a. Also S.a^p = 0, 34I-J xmEMATics. 197 and therefore the distance of any point in the path from the point — ^/3a~i/3 is equal to its distance from the line whose equation is Thus we recognise the focus and directrix property. 340.] That the moving point may reach a point y we must Have, for some real value of t. Now suppose Ty3, the velocity of projection, to be given =v, and, for shortness, write ot for Uj3. Then y = ^i^+viT^. Since Tzr = 1, we have («2 _ Say) i^ + Ty'^ = 0, The values of t'^ are real if {v^-Say^-Ta^Ty'^ is positive. Now, as TaTy is never less than Say, it is evident that v^ — Say must always be positive if the roots are possible. Hence, when they are possible, both values of i^ are positive. Thus we hscfefoiir values of t which satisfy the conditions, and it is easy to see that since, disregarding the signs, they are equal two and two, each pair refer to the same path, but described in ojaposite directions between the origin and the extremity of y. There are therefore, if any, in general two parabolas which satisfy the conditions. The directions of projection are (of course) given by the corresponding values of ct. 341.] The envelop of all the trajectories possible with a given velocity, evidently corresponds to {v^-Sayf-Ta''Ty^ = Q, for then y is the vector of intersection of two indefinitely close paths in the same vertical plane. Now v^ - Say = TaTy is evidently the equation of a paraboloid of revolution of which the origin is the focus, the axis parallel to a, and the directrix plane at a distance ^r- • la All the ordinary problems connected with parabolic motion are easily solved by means of the above formulae. Some, however, are even more easily treated by assuming a horizontal unit-vector in 198 • QUATERNIONS. [342. the plane of motion^ and expressing y3 in terms of it and a. But this must be left to the student. 342.] For acceleration directed to or from a fixed jaoint, we have, taking that point as origin, and putting P for the magnitude of the central acceleration, P =PUp. Whence, at once, f^pp = 0. Integrating, Fpp = y = a constant vector. The interpretation of this simple formula is — first, p and p are in a plane perpendicular to y, hence the path is in a plane (of course passing through the origin) ; second, the area of the triangle, two of whose sides are p and p is constant. [It is scarcely possible to imagine that a more simple proof than this can be given of the fundamental facts, that a central orbit is a plane curve, and that equal areas are described by the radius vector in equal times.J 343.] When the law of acceleration to or from the origin is that of the inverse square of the distance, we have p_ M Tp"' where p. is negative if the acceleration be directed to the origin. Hence p = ^ . The following beautiful method of integration is due to Hamilton. (See Chapter IV.) dJJp Vp.Vpp Up.y Generally, ^^ = - -^^ =--f^' , n .. dUp therefore py = —p. —j- , and py = e—pJJp, where e is a constant vector, perpendicular to y, because Sy'p = 0. Hence, in this case, we have for the hodograph, p = iy"^ — fji,Up.y~\ Of the two parts of this expression, which are both vectors, the first is constant, and the second is constant in length. Hence the locus of the extremity of p is a circle in a plane perpendicular to y (i.e. parallel to the plane of the orbit), whose radius is ^ > and whose centre is at the extremity of the vector ey""^. [This equation contains the whole theory of the Circular Hodo- 345-] KINEMATICS. 199 graph. Its consequences are developed at length in Hamilton's Wem,ents.'\ 344. J We may write the equations of this circle in the form y(p-ey-^) = Yy' (a sphere), and /Syp = (a plane through the origin, and through the centre of the sphere). The equation of the orbit is found by operating by Y.p upon that of the hodograph. We thus obtain y = r.pey-i + ^y/Dy-i, or y2 =Sip + ix.Tp, or txTp = Se{y^e-'^-p)-, in which last form we at once recognise the focus and directrix property. This is in fact the equation of a conicoid of revolution about its principal axis (e), and the origin is one of the foci. The orbit is found by combining it with the equation of its plane, Syp = 0. We see at once that y^ e^^ is the vector distance of the directrix . . . Te from the focus ; and similarly that the eccentricity is — j and the . -2My^ '' maior axis — = =- • 345.] To take a simpler case : let the acceleration vary as the dis- tance from the origin. Then p = ±m^p, the upper or lower sign being used according as the acceleration is from or to the centre. This is (^ + «.2)p = 0. Hence p = ae'"«+i3£-™'i or p = a cos mt + fi sin mt, where a and j3 are arbitrary, but constant, vectors; and e is the base of Napier's logarithms. The first is the equation of a hyperbola (§ 31, ^) of which a and ft are the directions of the asymptotes ; the second, that of an ellipse of which a and ft are semi-conjugate diameters. Since p == m {as'^ — fts'""} , or = m {—a sin mt + ft cos mt}, the hodograph is again a hyperbola or ellipse. But in the first case it is, if we neglect the change of dimensions indicated by the 200 QUATERNIOKS. [346. scalar factor m, conjugate to the orbit ; in the case of the ellipse it is similar and similarly situated. 346.] Again, let the acceleration he as the inverse third power of the distance, we have aUp Of course, we have, as usual, Vpp = y. Also, operating by S.p, ... (xSpp of which the integral is u the equation of energy. Again, Spp = -^ ■ Hence Spp + p'^ = C, or Spp = Ct, no constant being added if we reckon the time from the passage through the apse, where Spp = 0. We have, therefore, by a second integration, p^ = Cfi + C' (1) [To determine C", remark that pp = Ct + y, or pV = CH^-y'^. But p^p^ = Cp^—jj. (by the equation of energy), = CH^ + CC'-^, by(l). Hence CC'= ju-y^.] To complete the solution, we have, by § 133, where /3 is a unit-vector in the plane of the orbit. But r^ = - „ p p^ t — _y_. dt Hence ^°^~i" ^ "V i ^ ~ '^JCt^ + C The elimination of t between this equation and (1) gives Tp in terms of Up, or the required equation of the path. We may remark that if d be the ordinary polar angle in the orbit, T/o log^ = eUy. 348-] KINEMATICS, 201 Hence we have = —Ty I - W + G' \ and r^=-{Gt''-\-C'\ ) from which the ordinary ec[uations of Cotes' spirals can be at once found. [See Tait and Steele's Dynamics of a Particle, third edition, Appendix (A).] 347.] To find the conditions that a given curve may he the hodo- graph corresponding to a central orbit. If or be its vector, given as a function of the time, f^ndt is that of the orbit ; hence the requisite conditions are given by Tvjftsdt ■=■ y, where y is a constant vector. We may transform this into other shapes more resembling the Cartesian ones. Thus ^ FijfijTdt = 0, and * VzrfiiTdt+Vm'ST = 0. From the first f'^dt = x-ir, and therefore xYTsis = jf, or the curve \& plane. And m T^is + VisTs = ; or eliminating x, yViim = —(Fm-wY- Now if v' be the velocity in the hodographj 2if its radius of curva- ture, p' the perpendicular on the tangent ; this equation gives at once hv'= R'p"^, which agrees with known results. 348.] The equation of an epitrochoid or hypotrochoid, referred to the centre of the fixed circle, is evidently p = ai " a + M " a, where a is a unit-vector in the plane of the curve and i another perpendicular to it. Here o> and co^ are the angular velocities in the two circles, and t is the time elapsed since the tracing point and the centres of the two circles were in one straight line. Hence, for the length of an arc of such a curve, s —fT'pdt =fdt V { (o^a 2 + 2 tocoi a* cos (o) - (Oi) i! + ffli^ «2 } ^ = I dt ^y \{Q)a + <i)-J)f ±'i:u>a>^ah . ^ — o""^ 3' which is, of course, an elliptic function. 202 QUATERNIONS. [349- But when the curve becomes an epicycloid or a hypocycloid, coa+w^J = 0, and which can be expressed in finite terms, as was first shewn by Newton in the Principia. The hodograph is another curve of the same class, whose equa- tion is 2iat 2a)j< p = i{aa)i " a + bco-yi ^ a); and the acceleration is denoted in magnitude and direction by the vector iat 2M\t p = —au?i " a—ba\ i " a. Of course the equations of the common Cycloid and Trochoid may be easily deduced from these forms by making a indefinitely great and o) indefinitely small, but the product aa> finite ; and transferring the origin to the point p=. aa. 349.] Let i be the normal-vector to any plane. Let la- and p be the vectors of any two points in a rigid plate in contact with the plane. • After any small displacement of the rigid plate in its plane, let dm and dp be the increments of m and p. Then Sidm = 0, Sidp = ; and, since T^sr — p) is constant, S{-a-—p) idsr—dp) = 0. And we may evidently assume dp ■= a)i{p — t), dsT =: Q>J(or — t) ; where of course t is the vector of some point in the plane, to a rota- tion 0) about which the displacement is therefore equivalent. Eliminating it, we have d('ST — p) m = -^ —) •ST — p which gives to, and thence r is at once found. For any other point a- in the plane figure Sida- = 0, S{p—(t) {dp— da) = 0. Hence dp— da = aji«(p— o-). S{(T—'m){dijr — d(T) = 0. Hence dcr—dzr = oo^i^a—w). From which, at once, coj = w^ = co, and da- = (0? ((T— t), or this point also is displaced by a rotation a> about an axis through the extremity of r and parallel to i. 35 1 -J KINEMATICS. 203 • 350.] In the ease of a rigid body moving about a fixed point let OT, p, a- denote the Ysctors of any three points of the body ; the fixed point being origin. Then ■sr', p^, <j^ are constant^ and so are Sssp, Spar, and Sa-sr. After any small displacement we have, for tn- and p, Smdzs- = 0, ^ Spdp = 0, i (1) Szjdp + Spdzj = 0. ) Now these three equations are satisfied by «?sr =: VazT, dp = V^ap, where a is any vector whatever. But if dur and dp are given, then Vdsrdp ^ T^.FazrFap = aS.ap^. Operate by S.V^p, and remember (1), S^zrdp = S^pd^ = S^.apTn: Vdvrdp Vdpdxs ,. ^^^'^^ «= -s^ = -s^ ' ^') Now consider o-, Strdv = 0, \ Spdcr = —Strdp, V Sssda = — Sa-duT. ) da = Va<T satisfies them all, by (2), and we have thus the proposi- tion that ani/ small displacement of a rigid body abont a ^xed point is equivalent to a rotation. 351.1 To represent the rotation of a rigid body about a given awis^ ihrougli a given finite angle. Let a be a unit-vector in the direction of the axis, p the vector of any point in the body with reference to a fixed point in the axis, and 6 the angle of rotation. Then p = a''^Sap + a-^Vap, =■ — aSap — a Vap. The rotation leaves, of course, the first part unaffected, but the second evidently becomes — a ^ aVap, or — a Vap cos 6 + Vap sin 6. Hence p becomes pj z= — aSap — a Vap COS -f- Vap sin d, = (cos- + asm-jp(cos--asm-j. = a pa 204 QUATERNIONS. [352. 352.] Hence to compound two rotations about axes which meet, we may evidently write, as the effect of an additional rotation <^ about the unit-vector ;8, ^ _* Hence p^ = P' a" pa~ " p~' . If the /3-rotation had been first, and then the a-rotation, we should have had 1 ± _* _i. and the non-commutative property of quaternion multiplication shews that we have not, in general, P'i = ft- If a, fi, y be radii of the unit sphere to the corners of a spherical triangle whose angles are - > ^ . - > we know that U &i u ^ y" /3 " o "^ = — 1 . (Hamilton, Lectures, p. 267.) if. * Hence /3'o'=— y"'', -t ± and we may write P2 = y " py^^ or, successive rotations about radii to two corners of a spherical triangle, -and through angles double of those of the triangle, are equivalent to a. single rotation about the radius to the third corner, and through an angle double of the exterior angle of the triangle. Thus any number of successive _/?«Jfe rotations may be compounded into a single rotation about a definite axis. 353.] When the rotations are indefinitely small, the effect of one is, by § 351, p^ = f,-\-OiVap, and for the two, neglecting products of small quantities, p^ = p-\-(xTap+W^P, a and b representing the angles of rotation about the unit-vectors a and ^ respectively. But this is equivalent to P2 = p + r(aa-hb^)FV(aa + bi3)p, representing a rotation through an angle T{fi,a + b^), about the unit- vector TJ((xa + 6)3). Now the latter is the direction, and the former the length, of the diagonal of the parallelogram whose sides are (xa. and b/8. We may write these results more simply, by putting a for <ya, /3 for b/3, where a and ^ are now no longer unit-vectors, but repre- 355-1 KINEMATICS. 205 sent by their versors tlie axes, and by their tensors the angles (small)j of rotation. Thus pj^ = p+ Vap, P2 = p+Fap+ V^p, = p+Fia + p)p. 354.] The general theorem, of which a few preceding sections illustrate special cases, is this : By a rotation, about the axis of q, through double the angle of q, the quaternion r becomes the quaternion qrq~^. Its tensor and angle remain unchanged, its plane or axis alone varies. A glance at the figure is sufficient for . q the proof, if we note that of course T.qrq"^^ Tr, and therefore that we need consider the versor parts only. Let Q be the pole of q, A£=q, JJB' = q-\ WC' = r. Join C'A, and make AG = C'A. Join CB. Then CB is qrq-'^, its arc CB is evidently equal in length to that of r, B'C; and its plane (making the same angle with B'B that that of B'C does) has evidently been made to revolve about Q, the pole of q, through double the angle of q. If r be a vector, = p, then qpq"^ (which is also a vector) is the result of a rotation through double the angle of q about the axis of q. Hence, as Hamilton has expressed it, if B represent a rigid system, or assemblage of vectors, qBq-<^ is its new position after rotating through double the angle of q about the axis of q. 355.] To compound such rotations, we have r.qBq'^.r^''- =rq.B.{rq)-^. To cause rotation through an angle ^-fold the double of the angle of q we write q^Bq-K To reverse the direction of this rotation write q~^BqK To translate the body B without rotation, each point of it moving through the vector a, we write a + B. To produce rotation of the translated body about the same axis, and through the same angle, as before, q{a + B)q-\ Had we rotated first, and then translated, we should have had a + qBq-'^. 206 QUATERNIONS. [356. The obvious discrepance between these last results might perhaps be useful to those who do not believe in the Moon's rotation, but to such men quaternions are unintelligible. 356.] Given the instantaneous axis in terms of the time, it is re- quired to find the single rotation which will bring the body from any initial position to its position at a given time. If a be the initial vector of a point of the body, ot the value of the same at time t, and q the required quaternion, we have ^ = i°r^ (1) Differentiating with respect to t, this gives ■ar = qaq~^—qaq''^qq~^, = 2r.{rqq-\qaq-^). But ■a^ = Vei!7 = V.eqaq~^. Hence, as qaq"^ may be any vector whatever in the displaced body, we must have e = 2 Tqq-^ (2) This result may be stated in even a simpler form than (2), for we have always, whatever quaternion q may be, dUq Vqq-^ = dt {Vq)- and, therefore, if we suppose the tensor of q, which may have any value whatever, to be a constant (unity, for instance), we may write (2) in the form eq = 2q (3) An immediate consequence, which will be of use to us later, is q.q-''eq = 2q (4) 357.J To express q in terms of the usual angles i/f, 6, ^. Here the vectors i, J, h in the original position of the body corre- •spond to OA, OB, 00, respectively, at time t. The transposition is ef- fected by — first, a rotation y]r about k ; second, a rotation 6 about the new position of the line originally coinciding with/; third, a rotation (^ about the final position of the line at first coinciding with k. Let i, j, k be taken as the initial directions of the three vectors which at time t terminate at A, B, C re- spectively. The rotation >/f about h has the operator t _i k''{ )k ''. 357-] KINEMATICS. 207 This converts y into r), where t -i- . 'tj = k''ji " = J COS yp—i sin \{r. The body next rotates about tj through an angle 9. This has the operator t _* It converts k into ^ * _^ 6 Q Q 6 OC = C= yfk-q "= (cos- +'?sin-)/4(cos-— jjsin-) = ^cos04sind(jcos\/f + ysin\/f). The body now turns through the angle (p about C the operator being * _* Hence = (eos - + C sin -) (cos - + j? sin -) (cos| + k sin |) = [cos~ + Csm-) cos-cos^ + Acos-sin|^ . -Jf , . . . , . 6 . if ,. . . ~[ + sin-cos^(_;cosT//— »sini|f) + sin-sin^(«eosi/f+_;smi/f) a 2 2 2 _J / <^..^sr e ^// . . e . f . . e ^ir , ■ ■^i = (cos-+Csin-) cos-cos^ — »sm-sin-i^ + » sin -cos- + /ecos-sin— \ 2 ^ 2''L2 2 2 2-' 2 2 2 2j 4> ^ . (b . 6 . ■Jf . „ = cos — cos -cos— + sm — sin-sin— sm^cosiir 222222 ^ . A . e ^ ■ „ . , .</> e . ■jf — sm— sm-eos — sin5sm\/f— sm — cos- sin — cos ^ 2 2 2 ^ 2 2 2 .^ (j) . 6 . -f . <j) 6 ^ . „ + «( — cos — sin - sin — + sm — cos - cos — sm 6 cos <|f V 2 222 2'2 ^ . d) . 6 -Jf „ . (b e . \1/ . „ . N — sm — sm - cos ^ cos 9 + sm — cos - sin — sm sin i/j ) • 2 2 2 2 2 2 ^■' .f (b . d lif . (b e ^ . „ . + f I cos — sm - eos — + sm — cos - eos — sm a sm \|f ■^^222222 ^ . <b . e . f „ . <f) . -^ . „ ,\ — sm — sin - sm j- cos 0— sin — cos - sin sm 6 eos i/f ) 2 2 2 2 2 2 ^>' 7/0 . yj/ . <l) if a + ^ I cos — cos - sin — + sm ^ cos - cos — cos 9 V2 2 '2 2 2 2 .rf).e.>|f.^., . (h . ir . „ x + sm5-sin-sin — sm0sinvf+ sm— sin- cos— sm9cos\/r) 222 ^222 ^/ rf) + i/f 5 . . (b—yif . . (b—yJf . , . <b + ^lt 6 = cos cos - + « sm sm - +_; cos sin- + a sm -^^ — 2: cos- > which is, of course, essentially unsyuimetrical. 208 QUATERNIONS. [358. 358.] To find the usual equations connecting \j/, 6, (p with the an- gular velocities about three rectangular axes fixed in the body. Having tlie value of q in last section in terms of the three angles, it may be useful to employ it, in conjunction with equation (3) of § 356, partly as a verification of that equation. Of course, this is an exceedingly roundabout process, and does not in the least re- semble the simple one which is immediately suggested by qua- ternions. We have 2q=. eq= {<a^OJ + oo20£ + <agOC} q, ■whence ^i~^4 = S~^ {o)iOA + ai^ OB + co^ 00} q, or 2q = q{ia)i +ju>2 + ka^). This breaks up into the four (equivalent to three independent) equations 2 -7; ( cos ^^-—-!- cos-) dt^ 2 2> . d)—-J/ . 6 d> — -Jf . e . (b + \lf e = — CO, sm — — -^ sin - — o), cos - — — sin co„ sm - — — cos - j ^2 2^2 2^2 2 2-T:(sin ^„ ^ sin-) dt V 2 2'' (t> + \l/ e . (b + xir 6 (h — ylr . = 0), cos — - — cos- — coosm — — ^eos- + a),cos-^^ — i-sm-, 1 2 2 ^ 2 2 ^ 2 2 ^d ^ (b — \lr . 0\ 2 — (cos „ sin-l dt\ 2 2' . (t> + ^ (b + ylf . tb — yS, . = Q)j sin cos - + i»2 cos cos ;r — 0)3 Sin — — ~ sm - J 2 ^ ( sin ^ cos - ) dt^ 2 2' (b—-d/ . . <b — ^ . (h + y], = —6)1 cos sin- + 6)2 sin — —^ sm- + a>3C0S^!^ — 31 cos-- Prom the second and third eliminate 0— x/^, and we get by in- spection ^ a / ■ i , ,^ ^ ^ COS - . = (uj sm <p + 0)2 cos (^) cos - > or ^ = Wj sin (/) + Wg cos (/). (1) Similarly, by eliminating between the same two equations, . 0,- -.s ■ ^ ^ ^ • ^ sm — ((^ — \\r) = cog sin — + iBi cos 9 cos — — a>^ sm <p cos— • 2 i £1 2 And from the first and last of the group of four ■ ■ . . . cos-{(l> + \j/) = 0)3 cos- — WjCosc^ sm- + Wg sm <^ sm - • 2 2 i ^ 359-] KINEMATICS. 209 These last two equations give <j) + \jfCOsd = 0)3 (2) <p cos 6 + \jf = ( — w^ cos (^ + 0)2 sin (^) sin + 0)3 cos0. From the last two we have ■\jr sin 6 =— ctfj 008(^ + 0)2 sin (^ (3) (1), (2), (3) are the forms in which the equations are usually given. 359.] To deduce eiepressions for the direciion-coaines of a set 0/ rectangular axes in any position in terms of rational functions of three Let a, yS, y be unit- vectors in the directions of these axes. Let c[ be, as in § 356, the requisite quaternion operator for turning the coordinate axes into the position of this rectangular system. Then q^ = w + xi-^yj-^zh, where, as in § 356, we may write 1 = W^+iB^+y^+^2. Then we have (f^ =■ w—xi—yj-\-zk, and therefore a = qiq"^ = {wi—x—yk + zf){w—xi—yj—z}c) = [ie^ +x^ —y'^ —z^)i+ 2 {wz + xy)J +2(xz—toy)k, where the coefficients of i, J, k are the direction-eosines of a as required. A similar process gives by inspection those of ^ and y. As given by Cayley*, after Rodrigues, they have a slightly different and somewhat less simple form — to which, however, they are easily reduced by putting _'^_.5'_'^_ ^ \ jJ. V ^i The geometrical interpretation of either set is obvious from the nature of quaternions. For (taking Cayley's notation) if be the angle of rotation : cos^ cosy, cos h, the direction-cosines of the axis, we have 6 6 q = w + xi+yj + zJc = cos- + sin- (i cos/ +/ cosy + /i cos ^), Q SO that w — cos - > X = sm-cos/, a . y = sm-cosy; • ^ i z = sm - cos n. 2 * Camb. and Bub. Math. Journal. Vol. i. (1846.) 210 QUATERNIONS. [360. From these we pass at once to Rodrigues' subsidiary formulae, K = -5 = sec^ - . w^ 2 X = — = tan - cos/, &c. = &c. 360.J By the definition of Homogeneous Strain, it is evident that if we take any three (non-eoplanar) unit-vectors a, /3, y in an un- strained mass, they become after the strain other vectors, not neces- sarily unit- vectors, a^, ySj^, y^. Hence any other given vector, which of course may be thus ex- pressed, p =i xa + yfi -\- zy, becomes Pi = c(ia-^-^y^^->r zy-^, and is therefore known if a^, j3^, yj be given. More precisely pS.afiy = aS.j3yp + j3S.yap + yS.al3p becomes piS.a^y = (ppS.a^y = aj^S.^yp + ^-j^S.yap + yiS.a^p. Thus the properties of cf), as in Chapter V, enable us to study with great simplicity strains or displacements in a solid or liquid. For instance^ to find a vector whose direction is unchanged hy the strain, is to solve the equation Yp^p = 0, or <^p = gp, where ^ is a scalar unknown. [This vector equation is equivalent to three simple equations, and contains only three unknown quantities ; viz. two for the direction of p (the tensor does not enter, or, rather, is a factor of each side), and the unknown ^.] We have seen that every such equation leads to a cubic in g which may be written g^—m^g'^ + m^g—m = 0, where ««2i ^u ''"■ ^'"^ scalars depending in a known manner on the constant vectors involved in 0. This must have one real root, and may have three. 361. J For simplicity let us assume that a, /3, y form a rectangular system, then we may operate by S.a, S.^, and S.y; and thus at once obtain the equation for g, in the form 0... (1) Saoj^ 4 g. Sa^i, Say^ S^a^, Sl3^i+ff, S^y, Syai, SyPi, ^yyi+9 362.J KINEMATICS. 211 To reduce this we have 'S'aoi, Sa^i, Say^^ S^a^, Sj3p^, Sl3y^ Sya^, Sy^i, Syy^ 1 S^aa^ + S^pa^ + S^ya,, ^Saa^SajS^, Sya^, Syl3i, which, if the mass be rigid, becomes successively ^Saa^Sayi Sl3y, Syy^ Saa^ \Sypj^, mi = s^mMyri-7iSyPi) Thus the equation becomes - 1 -ff{Saa^ + aS^^i + Syy^) +g^ {Saa^ + Sp^^ + Syy^) +g^ = 0, {g-^){9^+9{^+Saa^ + Spp^ + 8yy;)+l) =0. 362.] If we take Tp :=G we consider a portion of the mass initially spherical. This becomes of course or n- C, an ellipsoid, in the strained state of the body. Or if we consider a portion which is spherical after the strain, i. e Tp^ = C, its initial form was T^p = C, another ellipsoid. The relation between these ellipsoids is obvious from their equations. (See § 311.) In either case the axes of the ellipsoid correspond to a rectangular set of three diameters of the sphere (§ 257). But we must care- fully separate the cases in which these corresponding lines in the two surfaces are, and are not, coincident. For, in the former case there is jmre strain, in the latter the strain is accompanied by ro- tation. Here w6 have at once the distinction pointed out by Stokes* and Helmholtzf between the cases of fluid motion in which there is, or is not, a velocity-potential. In ordinary fluid motion the distortion is of the nature of a pure strain, i.e. is differ- entially non-rotational ; while in vortex motion it is essentially ac- companied by rotation. But the resultant of two pure strains is generally a strain accompanied by rotation. The question before us beautifully illustrates the properties of the linear and vector function. * Cambridge Phil Trans. 1845. + Crelle, vol. Iv. 1857. See also Phil Mag. (Supplement) June 1867. P 2 212 QUATEENIONS. [363. 363.] To find the criterion of a pure strain. Take a, p, y now as unit-vectors parallel to the axes of the strain-ellipsoid, they become after the strain a a, bj3, cy. Hence p, = (pp ——aaSap—b^S^p — cySyp. And we have, for the criterion of a pure strain, the property of the function <\>, that it is self-conjugate, i. e. Sp(fi<T = S(T<pp. 364.J Two pure strains, in succession, generally give a strain ac- companied hy rotation. For if <p, \jf represent the strains, since they are pure we have Sp^a- = Sai^tp, ^ But for the compound strain we have Pi = XP = ^^P> and we have not generally Spx<T = Saxp. For 8p^<ji<T = Sa-(j)\jfp, by (1), and i/?0 is not generally the same as (f)\j/. (See Ex. 7 to Chapter V.) 365.] The simplicity of this view of the question leads us to suppose that we may easily separate the purs strain from the rotation in any case, and exhibit the corresponding functions. When the linear and vector function expressing a strain is self- conjugate the strain is pure. When not self-conjugate, it may be broken up into pure and rotational parts in various ways (analogous to the separation of a quaternion into the sum of a scalar and a vector part, or into the product of a tensor and a versor part), of which two are particularly noticeable. Denoting by a bar a self- conjugate function, we have thus either €l> = qw{ )q-\ or 4, = ^^.q{ )q-\ where e is a vector, and q a quaternion (which 'may obviously be regarded as a mere versor). That this is possible is seen from the fact that (j) invofves nine independent constants, while ^ and ct each involve six, and e and q each three. If </>' be the function conjugate to <f>, we have <j>'=f-F.e{ ), 60 that 2\}r = <p + <f)', and 2r.i{ ) = 0-()i', which completely determine the first decomposition. This is, of 365.] KINEMATICS. 213 course, perfectly well known in quaternions, but it does not seem to have been noticed as a theorem in the kinematics of strains that there is always one, and but one, mode of resolving a strain into the geometrical composition of the separate effects of (1) a pure strain, and (2) a rotation accompanied by uniform dilatation perpendicular to its axis, the dilatation being measured by (sec. 0—\) where Q is the angle of rotation. In the second form (whose solution does not appear to have been attempted), we have <t> = i^{ ) r\ where the pure strain precedes the rotation, and from this 0'=:^.j-i( )q, or in the conjugate strain the rotation (reversed) is followed by the pure strain. From these and OT is to be found by the solution of a biquadratic equation*. It is evident, indeed, from the identical equation S.CT<p'(l)p = S.p(j/(I><T that the operator ^'^ is self-conjugate. In the same way <^<^'( )=q^^{s-H )q)q-\ or §■-! {4,<t)'p) q = ^^ iq-'^pq) = ¥^ [q'^pq), which shew the relations between ^<^', <^'0, and q. To determine q we have <t>p-q = q^P * Suppose the cubic in ct to be ra^' + 3^"" + gr, TO^ + 32 = 0, write 0; for ^'<j> in the given equation, and by its help this may be written as (W + sf)a) + jriW= + g'2 = = w'(o) + g',)+araj + 5f2. Eliminating 5=, we have <"' + (2?, -ff") oi' + {g,''-2gg,)o,-g^ = 0. This must agree with the (known) cubic in ai, 0^ -i- mar' + m^a + ma=0, suppose, so that by comparison of coefficients we have so that g, is known, and g= ' • 2 -v/— ma where 2^. = m-(^^^ The values of the quantities g being found, w is given in terms of <u by the equation above. (Proc. B. S. E., 1870-71-) 214 QUATERNIONS. [366. whatever be p, so that S.Fq{<f)—m)p= 0, or S.p{<^'-^)Fq = Q, which gives {'¥~ ^) ^i = 0> The former equation gives evidently whatever be o and /3 ; and the rest of the solution follows at once. A similar process gives us the solution when the rotation precedes the pure strain. 366.] In general, if Pi = (jyp = —CiSap—^j^S^p—y-^Syp, the angle between any two lines, say p and a; becomes in the altered state of the body cos-^ {-S.U<l)pU<l><T). The plane *Sfp = becomes (witji the notation of § 144) SCpi = = SC<l>p = Sp<l>'C Hence the angle between the planes SCp = 0, and Srjp = 0, which is cos~^(—iS.UCUri), becomes cos-^{-S.U<l>'CU<l,'ri). The locus of lines equally elongated is, of course, T^Up = e, or T<i,p = eTp, a cone of the second order. 367.] In the case of a Simple Shear, we have, obviously, Pi = <i>P = p + fiSap, where Sa^ =0. The vectors which are unaltered in length are given by Tp^ = Tp, or 2 S^pSap + l3^S^ap = 0, which breaks up into S. ap = 0, and Sp{2fi + fi^a) = 0. The intersection of this plane with the plane of a, /3 is perpen- dicular to 2/3 + /3*a. Let it be a + a? /3, then -S.(2/3 + y32a)(a + a;/3) = 0, i.e. 2a! — 1 = 0. Hence the intersection required is 368.] KINEMATICS. 215 <^ For the axes of the strain, one is of course aj3, and the others are found by making TcjyJJp a maximum and minimum. Let p = a + x^, then pi= (pp = a + xj3—l3, and -^ = max. or mm., Tp 1 gives x^—x+-^ = 0, from which the values of x are found. Also, as a verification, S.{a + XiP){a + X2l3) =—l + p.'^x^x^, and should be = 0. It is so, since, by the equation, _ 1 Again S{a + {x^-l)fi} {a-\-{x^-\)p} =-\+&^{x^x^-{x^ + x.,)+l}, which ought also to be zero. And, in fact, aj^ + ^g = 1 by the equa- tion ; so that this also is verified. 368.] We regret that our limits do not allow us to enter farther upon this very beautiful application. But it may be interesting here, especially for the consideration of any continuous displacements of the particles of a mass, to in- troduce another of the extraordinary instruments of analysis which Hamilton has invented. Part of what is now to be given has been anticipated in last Chapter, but for continuity we commence afresh. If Fp = C (1) be the equation of one of a system of surfaces, and if the differential of (l)be Svdp= 0, (2) v is a vector perpendicular to the surface, and its length is inversely proportional to the normal distance hetween two consecutive surfaces. In fact (2) shews that v is perpendicular to dp, which is any tangent vector, thus proving the first assertion. Also, since in passing to a proximate surface we may write Svbp = 8C, we see that F{p + v-^hC) = C + W. This proves the latter assertion. It is evident from the above that if (1) be an equipotential, or an isothermal, surface, —v represents in direction and magnitude the force at any point or the flux of heat. And we have seen (§ 317) that if . d . d -. d dx '' dy dz 216 QUATERNIONS. [369. d'' A^ d^ gmng v^=______, then V = VFp. This is due to Hamilton (Lectures on Quaternions, p. 611). 369.] From this it follows that the effect of the vector operation V, upon any scalar function of the vector of a point, is to produce the vector which represents in magnitude and direction the most rapid change in the value of the function. Let us next consider the effect of V upon a vector function as <^ = ii+Jv + ^C- We have at once -=-{g-$-S)-'(S-f)-^- and in this" semi-Cartesian form it is easy to see that : — If T represent a small vector displacement of a point situated at the extremity of the vector p (drawn from the origin) SV a- represents the consequent cubical compression of the group of points in the vicinity of that considered, and VVa represents twice the vector axis of rotation of the same group of points. Similarly 5. V= - (^^ +, i- + C^) = -D., or is equivalent to total differentiation in virtue of our having passed from one end to the other of the vector a. 370.] Suppose we fix our attention upon a group of points which, originally filled a small sphere about the extremity of p as centre, whose equation referred to that point is To3 = e (1) After displacement p becomes p + a-, and, by last section, p + a> becomes p + m + cr— (jSa)V)o-. Hence the vector of the new surface which encloses the group of points (drawn from the extremity of p + tr) is Q)i = oi — {8<i>V)(T (2) Hence o) is a homogeneous linear and vector function of w-^ ; or and therefore, ^7 (1)> ^^o)i = e, the equation of the new surface, which is evidently a central surface of the second order, and therefore, of course, an ellipsoid. We may solve (2) vsdth great ease by approximation, if we re- member that T^ is very small, and therefore that in the small term we may put <Bj for w ; i. e. omit squares of small quantities ; thus (o = <Bj + (Sa>jV)a: 372-] KINEMATICS. '217 371.] If the small- displacement of each point of a medium is in the direction of, and proportional to, the attraction exerted at that point hy any system of material masses, the displacement is effected without rotation. For \i Fp = C be the potential surface^ we have Sddp a complete differentia] ; i. e. in Cartesian coordinates ^dx + r]di/ + (dz is a differential of three independent variables. Hence the vector axis of rotation ^ ^^ g vanishes by the vanishing of each of its constituents, or r.Va- = 0. Conversely, if there he no rotation, the displacements are in the direction of and proportional to, the normal vectors to a series of surfaces. For 0=r.dpr.Vcr = (SdpV) a- - ^Sadp, where, in the last term, V acts on o- alone. Now, of the two terms on the right, the first is a complete differ- ential, since it may be written —Dcip(T, and therefore the remaining term must be so. Thus, in a distorted system, there is no compression if SVa- = 0, and no rotation if V.Va = ; and evidently merely transference if o- = a = a constant vector, which is one case of Vg- = q. In the important case of a- = eVFp there is evidently no rotation^ since . Vff = eV^Fp is evidently a scalar. In this case, then, there are only translation and compression, and the latter is at each point proportional to the density of a distribution of matter, which would give the potential Fp. For if r be such density, we have at once V^Fp = 4 7rr*. 372.] The Moment of Inertia of a body about a unit vector a as axis is evidently jfp = -■2m{rapf, where p is the vector of the portion m of the mass, and the origin of p is in the axis. « Proc. B. 8. K, 1862-3, 218 QUATERNIONS. [372. Hence if we take hTa = e^, we have, as locus of the extremity of a, Jfe* =—^m,{Japf = MSai^a (suppose), the momental ellipsoid. If ts be the vector of the centre of inertia, o- the vector of m with respect to it, we have p = ot + o- ; therefore MB =-^m{{ Va^f + ( Faaf } = -M{ Vamf + MSa<i,^a. Now, for principal axes, Jc is max., min., or max.-min., with the condition ^z = _ 1 . Thus we have Sa{'arFaz7—(f)ia) — 0, Saa = ; therefore — ^la + wFatiT = ^a = h^o. (by operating by So). Hence (<^-^-\-k'^-\-vs^)a = +cr<S'aOT (1) detei-mines the values of a, Ic^ being found from the equation <St!r(<^ + P + OT2)-lt!7 = 1 (2) Now the normal to AS(r(0 + P + OT2)-^(7 = 1, (3) at the point o- is ((/> + ^^ + ot^)"^ o-. But (3) passes through — sr, by (2), and there the normal is which, by (1), is parallel to one of the required values of a. Thus we prove Binet's theorem that the' principal axes at any point are normals to the three surfaces, eonfocal with the momental ellipsoid, which pass through that point. EXAMPLES TO CHAPTER X. 1. Form, from kinematical principles, the equation of the cycloid ; and employ it to prove the well-known elementary properties of the arc, tangent, radius of curvature, and evolute, of the curve. 2. Interpret, kinematically, the equation p = aU{pt-p), where /3 is a given vector, and a a given scalar. Shew that it represents a plane curve ; and give it in an in- tegrated form independent of t. EXAMPLES TO CHAPTER X. 219 3. If we write ct = ^i—p, the equation in (2) becomes /3 — ■nr = aUv7. Interpret this kinematically ; and find an integal. What is the nature of the step we have taken in transforming from the equation of (2) to that of the present question ? 4. The motion of a point in a plane being given, refer it to {a.) Fixed rectangular vectors in the plane. {b.) Rectangular vectors in the plane, revolving uniformly about a fixed point. (c.) Vectors, in the plane, revolving with different, but uni- form, angular velocities. {d.) The vector radius of a fixed circle, drawn to the point of contact of a tangent from the moving point. In each case translate the result into Cartesian coordinates. 5. Any point of a line of given length, whose extremities move in fixed lines in a given plane, describes an ellipse. Shew how to find the centre, and axes, of this ellipse j and the angular velocity about the centre of the ellipse of the tracing point when the describing line rotates uniformly. Transform this construction so as to shew that the ellipse is a hypotrochoid. 6. A point. A, moves uniformly round one circular section of a cone; find the angular velocity of the point, a, in which the generating line passing through A meets a subcontrary section about the centre of that section. 7. Solve, generally, the problem of finding the path by which a point will pass in the least time from one given point to another, the velocity at the point of space whose vector is p being expressed by the given scalar function y^. Take also the following particular cases : — (a.) fp=.a while Sap> 1, fp = h while Sap < 1 . {h.) fp = Sap. (c.) fp = -p^. (Tait, Trans. R. S. E., 1865.) 8. If, in the preceding question,//) be such a function of Tp that any one swiftest path is a circle, every other such path is a circle, and all paths diverging from one point converge accurately in another. (Maxwell, Gam. and Bub. Math. Journal, IX. p. 9.) 220 QUATERNIONS 9. Interpret, as results of the composition of successive conical rotations, the apparent truisms y fi a and "^i -1^=1. Kid y p o. (Hamilton, Lectures, p. 334.) 1 0. Interpret, in the same way, the quaternion operators } = (8s-')*(<f-')*(f»"')'. 1 1 . rind the axis and angle of rotation by which one given rect- angular set of unit-vectors a, fi, y is changed into another given set Oi, Pi, yj. 12. Shew that, if <f>p = p+ Vep, the linear and vector operation (^ denotes rotation about the vector e, together with uniform expansion in all directions perpendicular to it. Prove this also by forming the operator which produces the expansion without the rotation, and that producing the rotation without the expansion ; and finding their joint effect. 13. Express by quaternions the motion of a side of one right cone rolling uniformly upon another which is fixed, the vertices of the two being coincident. 14. Given the simultaneous angular velocities of a body about the principal axes through its centre of inertia, find the position of these axes in space at any assigned instant. 15. Find the linear and vector function, and also the quaternion operator, by which we may pass, in any simple crystal of the cubical system, from the normal to one given face to that to an- other. How can we use them to distinguish a series of faces be- longing to the same zone ? 16. Classify the simple forms of the cubical system by the properties of the linear and vector function, or of the quaternion operator. 17. Find the vector normal of a face which truncates symmetri- cally the edge formed by the intersection of two given faces. 18. Find the normals of a pair of faces symmetrically truncating the g^ven edge. EXAMPLES TO CHAPTER X. 221 19. Find the normal of a lace which is equally inclined to three given faces. 20. Shew that the rhombic dodecahedron may be derived from the cube, or from the octahedron, by truncation of the edges. 2 1 . Find the form whose faces replace, symmetrically^ the edges of the rhombic dodecahedron. ♦ 22. Shew how the two kinds of hemihedral forms are indicated by the quaternion expressions. 23. Shew that the cube may be produced by truncating the edges of the regular tetrahedron. 24. Point out the modifications in the auxiliary vector function required in passing to the pyramidal and prismatic systems re- spectively. 25. In the rhombohedral system the auxiliary quaternion operator assumes a singularly simple form. Give this form, and point out the results indicated by it. 26. Shew that if the hodograph be a circle, and the acceleration be directed to a fixed point ; the orbit must be a conic section, which is limited to being a circle if the acceleration follow any other law than that of gravity. 27. In the hodograph corresponding to accelerationy(Z') directed towards a fixed centre, the curvature is inversely as D^y^D). 28. If two circular hodographs, having a common chord, which passes through, or tends towards, a common centre of force, be cut by any two common orthogonals, the sum of the two times of hodo- graphically describing the two intercepted arcs (small or large) will be the same for the two hodographs. (Hamilton, Mements, p. 725.) 29. Employ the last theorem to prove, after Lambert, that the time of describing any arc of an elliptic orbit may be expressed in terms of the chord of the arc and the extreme radii vectores. 30. If $'( )s~^ be the operator which turns one set of rect- angular unit- vectors a, /3, y into another set oj, /3^, y^, shew that there are three equations of the form CHAPTER XI. PHYSICAL APPLICATIONS. 373.] We propose to conclude the work by giving a few in- stances of the ready appHcability of quaternions to questions of mathematical physics, upon which, even more than on the Geo- metrical or Kinematical applications, the real usefulness of the Calculus must mainly depend — except, of course, in the eyes of that section of mathematicians for whom Transversals and Anharmonic Pencils, &c. have a to us incomprehensible charm. Of course we cannot attempt to give examples in all branches of physics, nor even to carry very far our investigations in any one branch : this Chapter is not intended to teach Physics, but merely to shew by a few examples how expressly and naturally quaternions seem to be fitted for attacking the problems it presents. We commence with a few general theorems in Dynamics — the formation of the equations of equilibrium and motion of a rigid system, some properties of the central axis, and the motion of a solid about its centre of inertia. 374.J When any forces act on a rigid body, the force /3 at the point whose vector is a, &c., then, if the body be slightly displaced, so that a becomes a + 6 a, the whole work done is 28pba. This must vanish if the forces are such as to maintain -equilibrium. Henoe ike condition of equilibrium of a rigid body is 2 SjSha = 0. For a displacement of translation 8a is ani/ constant vector, hence 2/3 = (1) For a rotation-displacement, we have by § 350, e being the axis, and Ti being indefinitely small, 6a = Ft a. 377-J PHYSICAL APPLICATIONS. 223 and S/S.^Tea = S/S.fTa/S = S.eliFafi) = 0, whatever be e, hence 2 . Ta^ = (2) These equations, (1) and (2), are equivalent to the ordinary six equations of equihbrium. 375.] In general, for any set of forces, let 2/3 = /3i, 2.ra/3 = ai, it is required to find the points for which the couple a-^ has its axis coincident with the resultant force ^^. Let y be the vector of such a point. Then for it the axis of the couple is 2.F(a-y)^ = ai-ry^i, and by condition x^-^ = a^ — Fy/Sj . Operate by S^-^ ; therefore x^l ^ ^ai/3i, and Ty^i = a^ -ft-^^iA = -^Ja^^^-^, or y = ^«i/3r^+.5'i3i, a straight line (the Central Axis) parallel to the resultant force. 376.] To find the points about which the couple is least. Here T{a^— Vyl3j) = minimum. Therefore S. (a^— FyjSj) F^iy = 0, where y' is any vector whatever. It is useless to try y'= ^^, but we may put it in succession equal to a^ and Vai^^. Thus S.yr.0^ra^P^ = Q, and {ra^^yf-fi\S.yra^p^ = 0. Hence y = x Va^ /Sj + j^/Sj , and by operating with S.Va^^^, we get Pi or y= ra^Py-"^ +y/3i, the same locus as in last section. 377.] The couple vanishes if «i- ^7/8i = 0. This necessitates Sa^fi^ = 0, or the force must be in the plane of the couple. If this be the case, still the central axis. 224 QUATERNIONS. [378. 378.] To assign the values of forces £, i^, to act at «, ej, and be equivalent to the given system. Hence Fe^H- n^ifi^-i) = a^, and i = (e- ei)-i (a^ - Tei ^1) + a; (e - €1). Similarly for f^. The indefinite terms may be omitted, as they must evidently be equal and opposite. In fact they are any equal and opposite forces whatever acting in the line joining the given points. . 379.] For the motion of a rigid system, we have of course ^S{md—/3)ba = 0, by the general equation of Lagrange. Suppose the displacements 6a to correspond to a mere translation, then 8a is an^ constant vector, hence '2{md — 0) = 0, or, if ai be the vector of the centre of inertia, and therefore a^'Em = 'Ema, we have at once di'Sm — 2/3 = 0, and the centre of inertia moves as if the whole masa were concen- trated in it, and acted upon by all the applied forces. 380.] Again, let the displacements 8 a correspond to a rotation about an axis «, passing through the origin, then ba = Fea, it being assumed that Te is indefinitely small. Hence I,S.eFa{m'd—^) = 0, for all values of e, and therefore I,.Fa{md-0) = 0, which contains the three remaining ordinary equations of motion. Transfer the origin to the centre of inertia, i. e. put a = a^ + ot, then our equation becomes 2r(a, + in-) (jKiii + »««■— /3) = 0. Or, since 2»»ot = 0, 2 Fot (»» OT - y3) + Fai(ai 2 J»- 2/3) = 0. But aj2»»— 2/3 = 0, hence our equation is simply ^V'mimih-^) = 0. Now 2Fi!r/3 is the couple, about the centre of inertia, produced by the applied forces ; call it (/>, then ImFs^ii = <{) (1) 383-] PHYSICAL APPLICATIONS. 225 381 .] Integrating once, • I.mF'ST^ = y+/<f)di (2) Again, as the motion considered is relative to the centre of inertia, it must be of the nature of rotation about some axis, in general variable. Let e denote at once the direction of, and the angular velocity about, this axis. Then, evidently, •a = Vetss. Hence, the last equation may be written 'S.mzrYiTS = yJrf^dt. Operating by S.i, we get 2m{Fem)^ = Sey + Se/<f,dt (3) But, by operating directly by 2fSidt upon the equation (1), we get 2»?(reCT)2 =-h^ + 2fSi<i>dt (4) (2) and (4) contain the usual four integrals of the first order. 382.] When no forces act on the body, we have ^ = 0, and therefore '2,mw Few = y, (5) Imir^ = ■2miFi'!:Tf = —A^, (6) and, from (5) and (6), Sey =—Jfi (7) One interpretation of (6) is, that the kinetic energy of rotation remains unchanged : another is, that the vector e terminates in an ellipsoid whose centre is the origin, and which therefore assigns the angular velocity when the direction of the axis is given ; (7) shews that the extremity of the instantaneous axis is always in a plane fixed^in space. Also, by (5), (7) is the equation of the tangent plane to (6) at the extremity of the vector e. Hence the ellipsoid (6) rolls on the plane (7). From (5) and (6), we have at once, as an equation which e must satisfy, y2 2.^ ( FimY= —k^ (2.»8sr Fivrf. This belongs to a cone of the second degree fixed in the body. Thus all the ordinary results regarding the motion of a rigid body under the action of no forces, the centre of inertia being fixed, are deduced almost intuitively : and the only difficulties to be met with in more complex properties of such motion are those of integration, which are inherent to the subject, and appear whatever analytical method is employed. (Hamilton, Proc. B. I. A. 1848.) 383.] Let a be the initial position of ■nr, q the quaternion by which the body can be at one step transferred from its initial posi- tion to its position at time t. Then ra- = qaq~^ Q 226 QUATEEKIONS. \.3M- and Hamilton's equation (5) of last section becomes or ^.mq {aS.aq~^ tq—q~^(qa? } q'"^ = y. Let <^p = 'Si.m{a8ap—a?p), (1) where is a self-conjugate linear and vector function, whose con- stituent vectors are fixed in the body in its initial position. Then the previous equation may be written or </>(S'~^«S') = rVS'- For simplicity let us write r^'i = r),^ Then Hamilton's dynamical equation becomes simply 0'? = C. (3) 384.3 ^^ is ®^y *o s^^ what the new vectors r\ and ( represent. For we may write (2) in the form e = qm-\ \ (2') from which it is obvious that rj is that vector in the initial position of the body which, at time t, becomes the instantaneous axis in the moving body. When no forces act, y is constant, and f is the initial position of the vector which, at time t, is perpendicular to the invariable plane. 385.] The complete solution of the problem is contained in equa- tions (2), (3) above, and (4) of § 356*. Writing them again, we have qr)=M, (4) 7i = iC, (2) 0'? = f. (3) We have only to eliminate f and »;, and we get 2q = q<f>-^q-^yq), (5) in which q is now the only unknown ; y, if variable, being supposed known in terms of q and t. It is hardly conceivable that any simpler, or more easily interpretable, equation for q can be presented * To these it is unnecessary to add Z'g= constant, as this constancy of Tq is proved by the form of (4). For, had Tq been variable, there must have been a quaternion in the place of the vector i/. In &ct, ^(Tqr = 2S.qKq^{Tqf8n'=0. 387.] PHYSICAL APPLICATIONS. 227 until symbols are devised far more comprehensive in their meaning than any we yet have. 386.] Before enfering into considerations as to the integration of this equation, we may investigate some other consequences of the group of equations in § 385. Thus, for instance, differentiating (2), we have and, eliminating q by means of (4), yqri + 2yq = qt,C+2qC, whence C=yCn+ i~^ yq. ; which gives, in the case when no forces act, the forms t=yp^-H, (6) and (as C= ^^) <l>ri= — F.ri<t>ri (7) To each of these the term q~^ yq, or q~^ yjfq, must be added on the right, if forces act. 387.] It is now desirable to examine the formation of the fanc- tion <f). By its definition (1) we have <l>p = 2.M (aSap — a^p), = — 'S.maVap. Hence —Sp(Pp = 'S,.m{Trapf, so that — Sp<pp is the moment of inertia of the body about the vector p, multiplied by the square of the tensor of p. Thus the equation g^^p ^ _p^ evidently belongs to an ellipsoid, of which the radii-vectores are inversely as the square roots of the moments of inertia about them ; so thatj if i, j, k be taken as unit- vectors in the directions of its axes respectively, we have Si<j)i = — A, \ Sj<f>j=-BA (8) Sk<t)k = -C,) A, B, C, being the principal moments of inertia, Consequently 4>p = —{AiSip + £JSjp+ CkSip} (9) Thus the equation (7) for rj breaks up, if w^ put into the three following scalar equations Aa)i+ (C— 5)q)2C»3 = 0, j Sd}^ + {A — C) w^coj^ = 0, I C(02 + {B — A) o>^a>2 = 0, ) Q 2 228 QUATEENIONS. [388. which are the same as those of Euler. Only, it is to be understood that the equations just written are not primarily to be considered as equations of rotation. They rather expres* with reference to fixed axes in the initial position of the body, the motion of the extremity, toj, Ug, (1)3, of the vector corresponding to the instan- taneous axis in the moving body. If, however, we consider tOj, Wg, cog as standing for their values in terms of w, x,y, «: (§ 391 below), or any other coordinates employed to refer the body to fixed axes, they are the equations of motion. Similar remarks apply to the equation which determines f, for if we put f=i^^ + y^^ + ^^^^ (6) may be reduced to three scalar equations of the form ''^9'^^ = 0. 388.] Euler's equations in their usual form are easily deduced from what precedes. For, let whatever be p ; that is, let + represent with reference to the moving principal axes what ^ represents with reference to the principal axes in the initial position of the body, and we have <t.e = q^ (q-^ iq) q'^ = q<l> (n) q'^ = qiq-' =qr{C'l>-H)q-' = -qr{ri<t>ri)q-^ . =-V.qri<p{n)q-'^ = -r.qr,q-'^q(t){q-'^eq)q-^ which is the required expression. But perhaps the simplest mode of obtaining this equation is to start with Hamilton's unintegrated equation, which for the case of no forces is simply S.»«FisrOT = 0. But from ot =: Vezr we deduce «• ?= Fe^+ Vk-sr = ore^ — e<S«CT+ Vkvr, so that 2.«M(F'e«riS€OT — eCT^ + cr^eBr) = 0. If we look at equation (1), and remember that ^ differs from simply in having ot substituted for a, we see that this may be written Fe+e + ^e = 0, 390-] PHYSICAL APPLICATIONS. 229 the equation before obtained. The first mode of arriving at it has been given because it leads to an interesting set of transformations, for which reason we append other two. By (2) y = qCq-\ therefore = qq-'^.q^q-'^+q^q-'^—q^^q-'^q^'^, or q^q-^ = iV.yVqq-'^ = Fye. But, by the beginning of this section, and by (5) of § 382, this is again the equation lately proved. Perhaps, however, the following is neater. It occurs in Hamil- ton's Elements. By (5) of §382 +€ = y. Hence <t>e =—<}>«=— S.w(t3- Few + ot Fenr) = — 'Si.m'iiSesi = — F'.f'2.m'srSe'ST = - re4.e. 389.] However they are obtained, such equations as those of § 387 were shewn long ago by Euler to be integrable as follows. letting 2fm^<i,^mjt = s, we have j^^z =JQ^^ + (£- C) s, with other two equations of the same form. Hence 2dt=: - so that t is known in terms of s by an elliptic integral. Thus, finally, tj or f may be expressed in terms of i ; and in some of the succeeding investigations for q we shall suppose this to have been done. It is with this integration, or an equivalent onCj that most writers on the farther development of the subject have commenced their investigations. 390.] By § 381, y is evidently the vector moment of momentum of the rigid body ; and the kinetic energy is But Sey = S.q-^eqq~^yq = SrjC so that when no forces act SC(l>'H=Sr]<l>r,=-AK But, by (2), we have also TC=Ty, or T<f>r, = Ty, so that we have, for the equations of the cones described in the 230 QUATERNIONS. [39 1. initial position of the body by rj and t, that is, for the cones de- scribed in the moving body by the instantaneous axis and by the perpendicular to the invariable plane, This is on the supposition that y and & are constants. If forces act, these quantities are functions of t, and the equations of the cones then described in the body must be found by eliminating t between the respective equations. The final results to which such a process will lead must, of course, depend entirely upon the way in which t is involved in these equations, and therefore no general statement on the subject can be made. 391.] Recurring to our equations for the determination of q, and taking first the case of no forces, we see that, if we assume tj to have been found (as in § 389) by means of elliptic integrals, we have to solve the equation „ .^ that is, we have to integrate a system of four other difiPerential equations harder than the first. Putting, as in § 3 8 7, n = icOj^ +j\ + kw^ , where Wj, Wg, W3 are supposed to be known functions of t, and q = w+ico + jy + kz, ... , . \ ,, dm dx du dz this system IS -di = ^ = y = Y ~ 'Z' * To get an idea of the nature of this equation, let us integrate it on the supposi- tion that ij is a constcmt vector. By differentiation and substitution, we get Hence „_ «,.«=. ^^ * j_ n =i- ^^ t g= ^icos — « + QsSin^ t. Substituting in the given equation we have 2^ C^ e, sin 2l e +& cos ^ «) = («, cos ^ «+ e, sin ^ ^j-J- Hence Tiy.Ga = Q, 1;, which are virtually the same equation, and thus. And the interpretation of 2 ( ) q~^ will obviously then be a rotation about ij through the angle tTrj, together with any other arbitrary rotation whatever. Thus any posi- tion whatever may be taken as the initial one of the body, and Q, ( ) Q,-» brings it to its required position at time < = 0. 3 9 2. J PHYSICAL APPLICATIONS. 231 m where ^= — <o,a;— Wgy — ojj^, X= Wj^W + tBg^ — a^i^, ^= w^w + a^se — ooi^l or, as suggested by Cayley to bring out the skew symmetry, X= . (ja^y — oi^z + ai-^w, T=.—m^x . + a-j^z + (o^w, Z ■=■ oj^a; — Wj^y . -^m^w, W ■=—<it>-^X — ai2,y — <«>3« . Here, of course^ one integral is w^ +(xi'^ +^^+z^ = constant. It may suffice thus to have alluded to a possible mode of solution, which, ^except for very simple values of ri, involves very great diffi- culties. The quaternion solution, when rj is of constant length and revolves uniformly in a right cone, will be given later. 392.] If, on the other hand, we eliminate t], we have to inte- grate S^~^ir^72)=^i' so that one integration theoretically suffices. But, in consequence of the present imperfect development of the quaternion calculus, the only known method of effecting this is to reduce the quaternion equation to a set of four ordinary differential equations of the first order. It may be interesting to form these equations. Put q = w+iai + jy + iz, Y = ia+Jb + ^o, then, by ordinary quaternion multiplication, we easily reduce the given equation to the following set : di d/w dx dy dz where W= — x'^—y3&—ze. or X= . yC— «13+wa, x= wa+^ffi— ^B r=— ««[; . +z^+wii, T= w'Q+z%—ui!<S; z= !JBi&—y% . +w(i::, and a = -J [a (w^— a;'' —y^ —z^) + 2a? {m + hy -\-ez) + 2w {bz—cy)'], 33 = -^ [5 {w^ —afi —y"^ —z^) + ly {ax + by + cz) + 2w {cx—az)'], a: = -^ [c {w^ —x^ —y^—z^) + 2z{ax+6y + cz) + 2w {ay-^bx)], 232 QUATERNIONS. [393« JF, X, Y, Zare thus homogeneom functions of w, x, y, z of the third degree. Perhaps the simplest way of obtaining these equations is to trans- late the group of § 385 into w, x, y, z at once, instead of using the equation from which f and r\ are eliminated. We thus see that ^ ^ *a+yi8 +/^ffi. One obvious integral of these equations ought to be vfi + x"^ +y^ +z^ = constant, which has been assumed all along. In fact, we see at once that wV+xX+yY+zZ= identically, which leads to the above integral. These equations appear to be worthy of attention, partly because of the homogeneity of the denominators W, X, T, Z, but particularly as they afford (what does not appear to have been sought) the means of solving this celebrated problem at one step, that is, without the previous integration of Euler's equations (§ 387). A set of equations identical with these^ but not in a homogeneous form (being expressed, in fact, in terms of k, \, |u, v of § 359, instead of 10, x,y, z), is given by Cayley {Gamb. and Bub. Math. Journal, vol. i. 1846), and completely integrated (in the sense of being re- duced to quadratures) by assuming Euler's equations to have been previously integrated. (Compare § 391.) Cayley's method may be even more easily applied to the above equations than to his own ; and I therefore leave this part of the development to the reader, who will at once see (as in § 391) that %, 38, ffi correspond to coi, Wg, tag of the rj type, § 387. 393.J It may be well to notice, in connection with the formulae for direction cosines in § 359 above, that we may write % = --j\a{:w'^-\-x^—y'''—z^)-^il{xy + 'wz)-\-'ic{pz—wy)'], 38 = -^\2a{xy — wz)-{h(vP-—x'^->ry'^—z''-)-\-1c{yz-\-wxy\, (!t = -p^[2a(xz + wy) + 2b {yz—wx) + c {w^ —x^ —y"^ + z^)']. These expressions may be considerably simplified by the usual assumption, that one of the fixed unit- vectors {i suppose) is perpen- dicular to the invariable plane, which amounts to assigning defi- nitely the initial position of one line in the body ; and which gives the relations 5—0 c = 396.] PHYSICAL APPLICATIOlSrS. . 233 • 394.] Wlieii forces act, y is variable, and the quantities a, h, c will in general involve all the variables w, x, y, z, t, so that the equations of last section become much more complicated. The type, however, remains the same if y involves t only ; if it involve q we must differentiate the equation, put in the form y=2q(l>{q-^q)q-^, and we thus easily obtain the differential equation of the second order ^ = iV.qct) (q-^ q) q-^ + 2 qcj) {F. q-^q) q-^ ; if we recollect that, because q~^q is a vector, we have 8.q-^=(3-^qf. Though remarkably simple, this formula, in the present state of the development of quaternions, must be looked on as intractable, except in certain very particular cases. 395.] Another mode of attacking the problem, at first sight entirely different from that in § 383, but in reality identical with it, is to seek the linear and vector function which expresses the Homogeneous Strain which the body must undergo to pass from its initial position to its position at time t. Let -ST = xfflj a being (as in § 383) the initial position of a vector of the body, ■ST its position at time t. In this case x i^ ^ linear and vector function. (See § 360.) Then, obviously, we have, ^-^ being the vector of some other point,, which had initially the value a^, Siss'ST^ = S.\a)(a.i = Saa^, (a particular case of which is T'ST = ^xa = Ta) and Fototj = J^-x^X^^i = x^"«i' These are necessary properties of the strain-function x, depending on the fact that in the present application the system is rigid. 396.] The kinematical equation CT = Few becomes Xa = F. exa (the function x being formed from x by the differentiation of its constituents with respect to t). Hamilton's kinetic equation S.warFera- = y, becomes 'Si.mxaF.exa = y. 234 QUATERNIONS. [39 7> This may be written 2.««(xaiS'.€xa— eo^) = V) or I,.m{aS.ax'f-x~^e.a^) = X~^Y> where x' is the conjugate of x- But, because '^•X'^X'h. = '^""u we have Saa^ = S.ax'xa^, whatever be a and a^, so that X = X ^• Hence 2.m{w8.ax~^e—x~^e.a^) = x~V, or, by §383, ^^-i^^^-iy^ 397.] Thus we have, as the analogues of the equations in §§ 383, 384, ^-1^ ^ ^^ x-V = C, and the former result x" = ^' «X° becomes X** =^'X'7X'* = X^**- This is our equation to determine X) V being supposed known. To find rj we may remark that <f>l = C, and C = X~V- But XX~^« = a. so that XX~^« + XX~'« = 0. Hence f=-X~^XX~V = -r.r,x-^Y=J'Cv=^-C4>-'C or </>^ = — Ftj^tj. These are the equations we obtained before. Having found rj from the last we have to find x from the condition X-^Xa=Fria. 398.] We might, however, have eliminated ?j so as to obtain an equation containing x a^lone, and corresponding to that of § 385. For this purpose we have jj = ,^-if= ^-^x"^y> so that, finally, X~^X'"' = ^- 't>~^ X~''>"»> or X~^ « = ^' X" ^ <'0~^X~^y' which may easily be formed from the preceding equation by putting X~^a for a, and attending to the value of x"^ given in last section. 40I.J PHYSICAL APPLICATIONS. 235 • 399.] We have given this process, though really a disguised form of that in §§ 383, 385, and though the final equations to which it leads are not quite so easily attacked in the way of integration as those there arrived at, mainly to shew how free a use we can make of symholic functional operators in quaternions without risk of error. It would be very interesting, however, to have the problem worked out afresh from this point of view by the help of the old analytical methods : as several new forms of long-known equations, and some useful transformations, would certainly be obtained. 400.] As a verification, let us now try to pass from the final equation, in x alone, of § 398 to that of § 385 in ^ alone. We have, obviously, OT = qa£r^ = X«. which gives the relation between q and x- [It shews, for instance, that, as yS.^Xa = -S.ax'A while 'S-zSxa = S.^qaq-'^ = S.aq-^^q, we have x'/3 = T'^Pi^ and therefore that xx'i^ = id'^Pi)^^ = i^, or x' = X~^j ^s above.] Difierentiating, we have qaq^'^—qaq'^qq'^ = x«- Hence X'^X" = S'~^?<*~"2^*? = 2r.r{q-'^q)a. Also ^~^X~V = ^"^(^^V?). so that the equation of § 398 becomes 2r.r{q-'^q)a= V. (^-^ {q-^7q) a, or, as a may have any value whatever, 2r.q'^q = ^-Hq-^yq), which, if we put Tq = constant as was originally assumed, may be written 2q = q<l>-\q-^yq), as in § 385. 401.] To form the equation for Precession and Nutation,. Let o- be the vector, from the centre of inertia of the earth, to a particle m of its mass : and let p be the vector of the disturbing body, whose mass is M. The vector-couple produced is evidently 236 QUATERNIONS. [4OI. = M^. mVap no farther terms being necessary, since =- is always small in the actual cases presented in nature. But, because o- is measured from the centre of inertia, S.?»o-= 0. Also, as in § 383, <^p = 2.«! {aScrp—tr'^p). Thus the vector-couple required is Referred to coordinates moving with the body, ^ becomes 4> as in § 388, and § 388 gives ♦e=y=3Jf/^ P^Pdt. Simplifying the value of <|> by assuming that the earth has two principal axes of equal moment of inertia, we have Bf—{A—B)aSaf = vector-constant + ZM{A—B) / ^g °^ dt. This gives Sat = const. = i2, whence e = — i2a -|- act, so that, finally, BVad-Aaa = ^{A-B)rap8ap. The most striking peculiarity of this equation is that Reform of the solution is entirely changed, not modified as in ordinary cases of disturbed motion, according to the nature of the value of p. Thus, when the right-hand side vanishes, we have an equation which, in the case of the earth, would represent the rolling of a cone fixed in the earth on one fixed in space, the angles of both being exceedingly small. If p be finite, but constant, we have a case nearly the same as that of a top, the axis on the whole revolving conically about p. 405. j PHYSICAL APPLICATIONS. 237 But if we assume the expr *sion p = r{Jeosmt + k sin mt)j (which represents a circular orbit described with uniform velocity,) a revolves on the whole conically about the vector i, perpendicular to the plane in which p lies. {Trans. B, 8. E., 1868-9.) 402.] To form the eq%iation of motion of a simple 'pendulum, taking account of the eartVs rotation. Let a be the vector (from the earth's centre) of the point of suspension, X its inclination to the plane of the equator, a the earth's radius drawn to that point ; and let the unit-vectors i, j, h be fixed in space, so that i is parallel to the earth's axis of rotation ; then, if m be the angular velocity of that rotation a = « p sin A + (/ cos 01^ + ^ sin ad) cos A] (1) This gives a = a o) ( —j sin tu^ + A cos mf) cos \ ^ = inYia ...(2) Similarly a = m Yia = — o)^ (a — ai sin A) (3) 403.] Let p be the vector of the bob m referred to the point of suspension, R the tension of the string, then if oj be the direction ofpuregravity m{d + p) =-mgUay-BUp, (4) which may be written rpd+rpp = ^ja,p (5) To this must be added, since r (the length of the string) is constant, Tp = r, (6), and the equations of motion are complete. 404.] These two equations (5) and (6) contain every possible case of the motion, from the most infinitesimal oscillations to the most rapid rotation about the point of suspension, so that it is necessary to adapt different processes for their solution in different cases. We take here only the ordinary Foucault case, to the degree of approximation usually given. 405.] Here we neglect terms involving m^. Thus we write a = 0, and we write a for Oj , as the difference depends upon the ellipticity of the earth. Also, attending to this, we have T p= — -a + i!T, (7) whereby (by (6)) xSoot = 0, (8) and terms of the order ot^ are neglected. 238 QUATERNIONS. [405. With (7), (5) becomes — — Vwss = — Foot ; a a so thatj if we write -■=•«?, (9) we have FaC* + w^ot) = (10) Now, the two vectors ai— asia\ and Via have, as is easily seen, equal tensors ; the first is parallel to the line drawn horizontally northwards from the point of suspension, the second horizontally eastwards. Let, therefore, w = «;(«»- o sin A) +j^ria, (11) which {x and y being very small) is consistent with (6). From this we have (employing (2) and (3), and omitting a?) •is = cb {ai— asinX) + yFia—xm ainXFia— yo) {a— ai sin \), a =z x{ai — aaiaK)+ifFia—2dia>BmKFia—2ya){a—aiBia\). With this (10) becomes Fa[ai(aJ— a sin \) + yFia—2xoi s\n\Fia—2ym{a—ai sin \) + n^x{ai—asm\)^n^yFia] = 0, or, if we note that F. a Fia = a{ai—a sin \), (^—x—2ya>smk—n,^x)aFia + {t/ — 2ii;a)8in.k + n'^y)a(ai—asm\) = 0. This gives at once x + n^x+ 2a>jfsm\ = 0, y+n^y—2a>xsin\ which are the equations usually obtained ; and of which the solution is as follows : — If we transform to a set of axes revolving in the horizontal plane at the point of suspension, the direction of motion being from the positive (northward) axis of x to the positive (eastward) axis of y, with angular velocity ii, so that a; = f cos Slt—r) sin Sit, ^ = f sin Qft + t) cos 12 1, and omit the terms in D? and in w 12 (a process justified by the results, see equation (15)), we have ({+«^0 cos Q,t-(ij + n^ri) sin Q.t-2^ {il—co sinX) = 0, ) So that, if we put il = oism\, (15) we have simply f +*^£ = 0, ) ij + n''r, = 0j ^^"^ the usual equations of elliptic motion about a centre of force in the centre of the ellipse. (Proc. E. S. K, 1869.) =::} <-> ';} <"' 407.-] PHYSICAL APPLICATIONS. 239 406.] To construct a reflecHkig surface from which rays, emitted from a point, shall after reflection diverge uniformly, hut horizontally. Using the ordinary property of a reflecting surface, we easily obtain the equation S.dp{^±^% = Q. By Hamilton's grand Theory of Systems of Bays, we at once write down the second form Tp—T(fi+aFap) = constant. The connection between these is easily shewn thus. Let ot and T be any two vectors whose tensors are equal, then whence, to a scalar factor ^re*, we have \i T + 'S!- = T Hence, putting w = C/'(/3 + aVap) and r = Up, we have from the first equation above S.dplUp+ Ui^ + aVap)'] = 0. But d(p + aFap) = aVadp =—dp—aSadp, and S.a(fi + a Vap) = .0, so that we have finally S.dpUp-S.d{^ + aFap)U{^ + arap) = 0, which is the differential of the second equation above. A curious particular case is a parabolic cylinder, as may be easily seen geo- metrically. The general surface has a parabolic section in the plane of a, y3 ; and a hyperbolic section in the plane of /3, a0. It is easy to see that this is but a single case of a large class of integrable scalar functions, whose general type is S.dp(^'p = 0, the equation of the reflecting surface ; while 8{<T—p)dcT — is the equation of the surface of the reflected wave : the integral of the former being, by the help of the latter, at once obtained in the form Tp + ^(a—p) = constant*. 407.] We next take Fresnel's Theory of DouMe Refraction, but * Proe. R. S. E., 1870-71. 240 QUATERNIONS. [408. merely for the purpose of shewing how quaternions simpHfy the processes required, and in no way to discuss the plausibility of the physical assumptions. Let tzT he the vector displacement of a portion of the ether, with the condition ^2 __i /j\ the force of restitution, on Fresnel's assumption, is tiflHSvar + b^jSj':!T + c^kSkin) = t<fm, using the notation of Chapter V. Here the function <^ is obviously self-conjugate, a^, b^, c^ are optical constants depending on the crystalline medium, and on the colour of the lightj and may be considered as given. Fresnel's second assumption is that the ether is incompressible, or that vibrations normal to a wave front are inadmissible. If, then, a be the unit normal to a plane wave in the crystal, we have of course a^=-\, (2) and Six's! = 0; (3) but, and in addition, we have ■s!~^ Vtz^Ts II a, or S.aTu^ = (4) This equation (4) is the embodiment of Fresnel's second assumption, but it may evidently be read as meaning, the normal to the front, the direction, of vibration, and that of the force of restitution are in one plane, 408.] Equations (3) and (4), if satisfied by -m, are also satisfied by Tsa, so that the plane (3) intersects the cone (4). in two lines at right angles to each, other. That is, for any given wave front there are two directions of vibration, and they are perpendicular to each other. 409.] The square of the normal velocity of propagation of a plane wave is proportional to the ratio of the resolved part of the force of restitution in the direction of vibration, to the amount of displace- ment, hence j;2 = S-as^Tn, Hence Fresnel's Wave-surface is the envelop of the plane Sap ^ i\/Sm<^, (5) with the conditions vt^ = — \, (1) a''=-l, (2) Sour =0, (3) S. aiJ7<l)'ar = (4) 409-] PHYSICAL APPLICATIONS. 241 Formidable as this problem appears, it is easy enough. From (3) and (4) we get at onee^ Henee^ operating by S. ct, — CO ^ — S'STcfyar = — v^. Therefore ((jb + »2) ^ = _ a^ac^^-, and S.a {(j) + v^)-^ a = (6) In passing, we may remark that tMs equation gives the normal velo- cities of the two rays whose fronts are perpendicular to a. In Cartesian coordinates it is the well-known equation P wfi rfi _ a^—v^ ■*" P3^2 + ^2~^ = °- By this elimination of or, our equations are reduced to S.a{(i> + v^)-'^a= 0, (6) V zzz-Sap, (5) a^ =-1 (2) They give at once, by § 309, {ct> + v^)-^a + vpSa{cj>-{-v^)-^a = ha. Operating by S.a we have v^Sa{<tj + v^)-^a = h. Substituting for h, and remarking that Sa{(t> + v^)-^a =-T^{(j> + t)2)-i a, because <^ is self-conjugate, we have /J . 2\-i va — p p^ + v^ This gives at once, by rearrangement, ^{(l> + v^)-^a = {<t>-p^)-Y Hence {<t>-p^)-^P = ^^^ ■ Operating by S.p on this equation we have Sp{<P-p')-^p = -l, (7) which is the required eqjflation. [It will be a good exercise for.the student to translate the last ten formulae into Cartesian coordinates. He will thus reproduce almost exactly the steps by which Archibald Smith * first arrived at a simple and symmetrical mode of .effecting the elimination. Yet, as we shall presently see, the above process is far from being the shortest and easiest to which quaternions conduct us.J * Cambridge Phil. Trans., 1835. 242 QUATEENIONS. [4IO. 410.] The Cartesian form of the equation (7) is not the usual one. It is, of course, aj2 yi g^ But write (7) .in the form and we have the usual expression a2^2 ^2 ,,2 „2,2 I 7,2 „2 T .!! ..•>. " ast quaternion eqi forms This last quaternion equation can also be put into either of the new or 2'(p-2-,^-i)-4p = 0. 411.] By applying the results of §§ 171, 172 we may introduce a multitude of new forms. We must confine ourselves to the most simple ; but the student may easily investigate others by a process precisely similar to that which follows. Writing the equation of the wave as where we have g = — p~^, we see that it may be changed to if mSp<f>p = ffkp^ ■=—h. Thus the new form is ^ Sp{(j)-^—mSp(l)p)-^p = (1) Here m = -^^^ , 8p^p = a^ap' + V^y"^ + c^z^, and the equation of the wave in Cartesian coordinates is, putting ' ^ + -...« ... = 0. 412.] By means of equation (1) of last section we may easily prove Pliicker's Theorem, The Wave-Surface is its own reciprocal with resjieci to the ellipsoid, whose equation is Sp^^p = —7— • 41 3-] PHYSICAL APPLICATIONS. 243 The equation of the plane of contact of tangents to this surface from the point whose vector is p is iSWd)* p = —, — The reciprocal of this platie, with respect to the unit-sphere about the origin, has therefore a vector cr where a = \/m,(ji^ p. Hence p = —t— (b~^a; and when this is substituted in the equation of the wave we have for the reciprocal (with respect to the unit-sphere) of the reciprocal of the wave with respect to the above ellipsoid, S.cr (^ - — Sacj)-^ 0-) 0- = 0. This differs from the equation (1) of last section solely in having (p~^ instead of (f>, and (consistently with this) — instead of m. Hence it represents the index-surface. The required reciprocal of the wave with reference to the ellipsoid is therefore the wave itself. 413.J Hamilton has given a remarkably simple investigation of the form of the equation of the wave-surface, in his Elements, p. 736, which the reader may consult with advantage. The following is essentially the same, but several steps of the process, which a skilled analyst would not require to write down, are retained for the benefit of the learner. Let %= — 1 (1) be the equation of any tangent plane to the wave^ i.e. of any wave- front. Then /u is the vector of wave-slowuess, and the normal velocity of propagation is therefore -=p- . Hence, if isr be the vector direction of displacement, ju~^«r is the effective component of the force of restitution. Hence, ^w denoting the whole force of re- stitution, we have ^'sr—pr^'oi || p., or -m II {4>—ijr^)-^p., and, as ss is in the plane of the wave-front, Sp.'d = 0, or SiJi.{(f)-p.-'')-^iJ. = (2) This is, in reality, equation (6) of § 409. It appears here, how- ever, as the equation of the Index-Surface, the polar reciprocal of E % 244 QUATERNIONS. [4 1 4. the wave with respect to a unit-sphere about the origin. Of course the optical part of the problem is now solved, all that remains being the geometrical process of § 3 1 1 . 414.] Equation (2) of last section may be at once transformed, by the process of § 410, into 5f.((^2-<|,-i)-V=i. Let us employ an auxiliary vector whence ij,= (jx'^—(J)-^)t (1) The equation now becomes Sh.t=1, (2) or, by (1), y?T-^-S!r4r'^T = 1 .- (3) Differentiating (3), subtract its half from the result obtained by operating with S.t on the differential of (1). The remainder is T'^Sixdn—STdjj. = 0. But we have also (§311) Spdix = 0, and therefore xp = jxt^—t, where a; is a scalar. This equation, with (2), shews that Stp = (4) Hence, operating on it by S.p, we have by (1) of last section xp^ = — r^, and therefore p~^ =— /x + r"^. This gives p~^ = ij,^ — t~'^. Substituting from these equations in (1) above, it becomes or r = ((^~^— p~^)~^p~^. Finally, we have for the required equation, by (4), ^p-i(<^-i_p-2)-ip-i = 0, or, by a transformation already employed, Sp{cf>-p^)-^p=-l. 415,] It may assist the student in the practice of quaternion analysis, which is our main object, if we give a few of these invest- igations by a somewhat varied process. Thus, in § 407, let us write as in § 168, aHSv^Jfl^jSj-ss^c^hSk^ = yxS/OT + Z/SW-yOT. We have, by the same processes as in § 407, S.VTaX'Si/t!r + S.'maix'Sk''!!T = 0. 4I7.J PHYSICAL APPLICATIONS, 245 (1) This may be written, so flr as the generating lines we require are concerned, since -sra is a vector. Or we may write S.[l,'V.'7T\'-S!a = = /S./yl'OTX'OTa. Equations (1) denote two cones of the second order which pass through the intersections of (3) and (4) of § 407. Hence their in- tersections are the directions of vibration. 416.] By (1) we have S.T!TX.''sraix'= 0. Hence ■nrX'tn-j a, \i.' are coplanar ; and, as tn- is perpendicular to a, it is equally inclined to Vk'a and Fix a. For, i£ L, M, A be the projections of k', f/, a on the unit sphere, £C the g-reat circle whose pole is A, we are to find for the projections of the values of w^ on the sphere points P and P', such that if LF be produced till Q may lie on the great circle AM. Hence, evidently, CP = PB, and C^F=rB; which proves the proposition, since the projections of Vk'a and Vj/a on the sphere are points b and c in BC, distant by quadrants from C and B respectively. 417.] Or thus, Svra = Q, S.srV.ak'-snx — 0, therefore as'sr = F. a K ak''as-ii, = - r. W/ -aSaF. W/x'. Hence {Sk'ix-a;) ot = (X' + aSak') ^/x'w + {/ + aSaf/) Sk'w. Operate by S.k', and we have (x + Sk'aSi/a) Sk'^ = [X'^ a^-S'^ X'a] -S/xV = Si/^T'^Fk'a. Hence by symmetry, ^''''^T^Fk'a=f^T^Fi/a, Sk'^a Sjjfz 246 QUATERNIONS. [4 1 8. "'' T7k'a - TFi/a - ' and as fco = 0, ^=U{Ur\'a±Ur,jfa). 418.] The optical interpretation of the common result of the last two sections is that the planes of polarization of the two rays whose wave-fronts are parallel, iisect the angles contained hy planes passing through the normal to the wave-front and the vectors (optic axes) A'j fx'. 419.] As in § 409, the normal velocity is given by v^ ^SsTCJysr = 2SX.''aSf/tsy-p'^^ = / + ; {T+8).r\'aF/a [This transformation, effected by means of the value of or in § 417, is left to the reader.] HencCj if w^, v^ be the velocities of the two waves whose normal is a, „2 _ ^.| ^ 2 T. r\'a r/a oc sin K'a sin ju'o. That is, the difference of the squares of the velocities of the two waves varies as the product of the sines of the angles between the normal to the wave-front and the optic axes (A', \j.'). 420.] We have, obviously. Hence v^=p'^^ {T± S). VK'a Ff/a. The equation of the index surface, for which Tp = -, Up = a, V is therefore 1 = -p'p^ + {T±S). Fx'p Fpfp. This will, of course, become the equation of the reciprocal of the index-surface, i.e. the wave-surface, if we put for the function ^ its reciprocal : i. e. if in the values of A', p.', p' we put - , y- , - for a, b, c respectively. We have then, and indeed it might have been deduced even more simply as a transformation of § 409 (7), \ = -pp^i;-{T±S).F\pFp.p, as another form of the equation of Fresnel's wave. 424-J PHYSICAL APPLICATIONS. 247 If we employ the i, k transformation of § 1 2 1, this may be written, as the student may easily prove, in the form 421. J We may now, in furtherance of our object, which is to give varied examples of quaternions, not complete treatment of any one subject, proceed to deduce some of the properties of the wave- surface from the diflFerent forms of its equation which we have given. 422.] Fresnel's construction of the wave hy points. From § 273 (4) we see at once that the lengths of the principal semidiameters of the central section of the ellipsoid Sp<^-^p = 1, by the plane Sap = 0, are determined by the equation 6'.a(<^-i-p-2)-ia=0. If these lengths be laid off along a, the central perpendicular to the cutting plane, their extremities lie on a surface for which a = Vp^ and Tp has values determined by the equation. Hence the equation of the locus is as in §§409, 414. ^P (r^-P'^V = 0, Of course the index-surface is derived from the reciprocal ellip- soid Sp>^p = 1 by the same construction. 423.] Again, in the equation \=-pp':^{T±S).YKpTp.p, suppose VKp = 0, or F/xp = 0, we obviously have U\ , Up. P = ±—7= or p = ±—=> vj) vp and there are therefore four singular points. To find the nature of the surface near these points put UK P = V^ + ^' where Tsr is very smallj and reject terms above the first order in Ttsr. The equation of the wave becomes, in the neighbourhood of the singular point, 2^35^^ + /S.OT r. X VXp. = ±T. TAot FX/x, which belongs to a cone of the second order. 424.] From the similarity of its equation to that of the wave, it 248 QUATERNIONS. [425. is obvious that the index-surface also has four conical cusps. As an infinite number of tangent planes can be drawn at such a point, the reciprocal surface must be capable of being touched by a plane at an infinite number of points ; so that the wave-surface has four tangent planes which touch it along ridges. To find their form, let us employ the last form of equation of the wave in § 420. If we put Trip=TrKp, (1) we have the equation of a cone of the second degree. It meets the wave at its intersections with the planes S{l-K)p=+{K^-i^) (2) Now the wave-surface is touched by these planes, because we cannot have the quantity on the first side of this equation greater in abso- lute magnitude than that on the second, so long as p satisfies the equation of the wave. That the curves of contact are circles appears at once firom (1) and (2), for they give in combination p2 = +5(t + K)p, (3) the equations of two spheres on which the curves in question are situated. The diameter of this circular ridge is [Simple as these processes are, the student will find on trial that the equation Sp{<f>~''-—p~^)~'^p = 0, gives the results quite as simply. For we have only to examine the eases in which — p"^ has the value of one of the roots of the symbolical cubic in (^"^. In the present case Tp = b is the only one which requires to be studied.] 425.] By § 41 3, we see that the auxiliary vector of the succeed- ing section, viz. is parallel to the direction of the force of restitution, 0in-. Hence, as Hamilton has shewn, the equation of the wave, in the form Srp = 0, (4) of §414, indicates that fJie direction of the force of restitution is perpendicular to the ray. Again, as for any one versor of a vector of the wave there are two values of the tensor, which are found from the equation 429-j PHYSICAL APPLICATIONS. 249 we see by § 422 that the lines of vibration for a given plane front are parallel to the axes of any section of the ellipsoid, S.p(t>-^p = 1 made hy a plane parallel to the front ; or to the tangents to the lines of curvature at a point where the tangent plane is parallel to the wave- front. 426.] Again, a curve which is drawn on the loave-surface so as to' touch at each point the corresponding line of vibration has Hence S(ppdp = 0, or Sp^p = C, so that such, curves are the intersections of the wave with a series of ellipsoids concentric with it. 427,] For curves cutting at right angles the lines. of vibration we have dp II Fp(j)-^ ((/)-! -p-^)-V \\rp{cj,-p^)-^p. Hence Spdp = 0, or Tp = C, so that the curves in question lie on concentric spheres. They are also spherical conies, because where Tp = C the equation of the wave becomes the equation of a cyclic cone, whose vertex is at the common centre of the sphere and the wave-surface, and which cuts them in their curve of intersection. (Quarterly Math. Journal, 1859.) 428.] As another example we take the case of the action of electric currents on one another or on magnets; and the mutual action of permanent magnets. A comparison between the processes we employ and those of Ampere {Theorie des Phenomenes Mectrodynamiques, ^c, many of which are well given by Murphy in his Electricity) will at once shew how much is gained in simplicity and directness by the use of quaternions. The same gain in simplicity will be noticed in the investigations of the mutual effects of permanent magnets, where the resultant forces and couples are at once introduced in their most natural and direct forms. 429.] Ampere's experimental laws may be stated as follows : I. Equal and opposite currents in the same conductor produce equal and opposite effects on other conductors : whence it follows 250 QUATERNIONS. [430. that an element of one current has no effect on an element of an- other which lies in the plane bisecting the former at right angles. II. The effect of a conductor bent or twisted in any manner is equivalent to that of a straight one, provided that the two are traversed by equal currents, and the former nearly coincides with the latter. III. No closed circuit can set in motion an element of 'a circular conductor about an axis through the centre of the circle and per- pendicular to its plane. IV. In similar systems traversed by equal currents the forces are equal. To these we add the assumption that the action between two elements of currents is in the straight line joining them : and two others, viz. that the effect of any element of a current on another is directly as the product of the strengths of the currents^ and of the lengths of the elements. 430.] Let there be two closed currents whose strengths are a and a^; let a, Oj be elements of these, a being the vector joining their middle points. Then the effect of a on oj must, when resolved along Oj, be a complete differential with respect to a (i.e. with respect to the three independent variables involved in a), since the total resolved effect of the closed circuit of which a' is an element is zero by III. Also by I, II, the effect is a function of Ta, Saa, Saa^, and 8a a^, since these are suflScient to resolve a and Oj into elements parallel and perpendicular to each other and to a. Hence the mutual effect is aa-JJaf{Ta, Saa, Saa^, Si/aj), and the resolved effect parallel to a^ is aiZj SUai TJaf. Also, that action and reaction may be equal in absolute magnitude, ymust be symmetrical in Sao! and Saa-^. Again, d (as differential of a) can enter only to the first power, and must appear in each term of/. Hence f^ASaa-^-\-^SaaSaa^. But, by .IV, this must be independent of the dimensions of the system. Hence J is of — 2 and ^ of — 4 dimensions in Ta. There- ^""■^ ^ {ASaa^Sda^ + BSaa'S^aa^} is a complete differential, with respect to a, if da = a. Let 433-] PHYSICAL APPLICATIONS. 251 where C is a constant deperaing on the units employed, therefore =.-=r; baa, 2Ta^ ~ Ta and the resolved effect Gaa^ S^aa^ Saa, „„ , , » o /o ^ "^ W^ IhF "^ 1 Ta Ta^ ^ ~ "i +^ 1^ = Caa^ „ y,^g {S. Vaa' Faa^-{-\ Saa'Saa^ . The factor in brackets is evidently proportional in the ordinary notation to sin 6 sin 6'cos ia — \ cos 6 cos 6'. 431.] Thus the whole force is Caa-^a , S^aa-^ _ Caa■^^a , S'^aa' as we should expect, d-^a being = a^. [This may easily be trans- formed into 2Caa,Ua which is the quaternion expression for Ampere's well-known form.] 432.] The whole effect on Oj of the closed circuit, of which a is an element, is therefore Cfeffj C a JSaa-^^ H f a J Saa^ 2 J Saa, Ta 3 between proper limits. As the integrated part is the same at both limits, the effect is ^•^^IF a I, a f^°-^' fdUa - V-^"^^' ''^''' ^=J'T^=J-^' and depends on the form of the closed circuit. 433.] This vector ^, which is of great importance in the whole theory of the effects of closed or indefinitely extended circuits, cor- responds to the line which is called by Ampere " direcfrice de V action electrodynamique" It has a definite value at each point of space, independent of the existence of any other current. Consider the circuit a polygon whose sides are indefinitely small; join its angular points with any assumed point, erect at the latter, perpendicular to the plane of each elementary triangle so formed, a (■> vector whose length is - > where to is the vertical angle of the tri- 252 QUATERNIONS. [434- angle and r the length of one of the containing sides ; the sum of such vectors is the " directrice" at the assumed point. 434.] The meve/orm of the result of § 432 shews at once that if the element Oj he turned about its middle point, the direction of the resultant action is confined to the plane whose normal is j3. Suppose that the element Oj is forced to remain perpendicular to some given vector 6, we have Soj^b = 0, and the whole action in its plane of motion is proportional to Tr.bFa^^. But r.bra^li=-a^S^b. Hence the action is evidently constant for all possible positions of a^ ; or The effect of any system of closed currents on an element of a con- ductor which is restricted to a given plane is {in that plane) independent of the direction of the element. 435.] Let the closed current be plane and very small. Let e (where Tt =■ 1 ) be its normal, and let y be the vector of any point within it (as the centre of inertia of its area) ; the middle point of oj being the origin of vectors. Let a = y + p; therefore a'= p, and .-/•^""-/• ^(y + P) / and P-J Ta?-J T(y + pY ^/r(..-,).'{l+^^-^ to a sufficient approximation. Now (between limits) fVpp'= 2Ae, where A is the area of the closed circuit. Also generally fVyp'Syp =^\{SypVyp^y7.yfVpp') = (between limits) AyVye. Hence for this case A , 3yFye>. ^=TyS{^' + -^) A ( 3y% x - Ty^\ '^ Ty"^ )' 436.] If, instead of one small plane closed current, there be a series of such, of equal area, disposed regularly in a tubular form, let X be the distance between two consecutive currents measured along the axis of the tube; then, putting y'= xs, we have for the whole effect of such a set of currents on a^ 438. j PHYSICAL APPLICATIONS. 253 g-^^«i V. fry J. ^y^yY' \ CAaa.^ Va■^y ,. , t -j. x = — - — - „ 3 (between proper limits). If the axis of the tubular arrangement be a closed curve this will evidently vanish. Hence a closed solenoid exerts no influence on an element of a conductor. The same is evidently true if the solenoid he indefinite in both directions. If the axis extend to infinity in one direction, and y^ be the vector of the other extremity, the effect is CAaa^ VoiVo and is thevefove perpendicular to the element and to the line joining it with the extremity of the solenoid. It is evidently inversely as Ty'% and directly as the sine of the angle contained letmeen the direction of the element and that of the line joining the latter with the extremity of the solenoid. It is also inversely as x, and therefore directly as the number of currents in a unit of the axis of the solenoid. 437.] To find the effect of the whole circuit whose element is Oj on the extremity of the solenoid, we must change the sign of the above and put a^ = y^; therefore the effect is _ CAaa^ r Vygyg 2x J Ty% ' an integral of the species considered in § 432 whose value is easily assigned in particular cases. 438.] Suppose the conductor to le straight, and indefinitely extended in both directions. Let ho be the vector perpendicular to it from the extremity of the canal, and let the conductor be || 77, where Td = Tri = 1 . Therefore yg = h6+yr} (where y is a scalar), TyoVo = A/J'rie, and the integral in § 436 is hr-qej J —CD -.=w. -00 {h-'+y^f h The whole effect is therefore xh and is thus perpendicular to the plane passing through the conductor and the extremity of the canal, and varies inversely as the distance of the latter from the conductor. 254 QUATERNIONS. [439- This is exactly the observed effect of an indefinite straight current on a magnetic pole, or particle of free ina;gnetism. 439.] Suppose the conductor to be circular, and the pole nearly in its axis. Let UPD be the conductor, A£ its axis, and C the pole ; £C perpendicular to A£, and small in comparison with AE = h the radius of the circle. Let AJB be Oji, where BC=hk, AP = h{jx + i wJ 'sm-" '•sm.-' Then cc/j/^ CP = y =. aii-\-bk—h{jx-\-ky). • [Fyy And the effect on C<x -^fy , ''6' {{h—by)i+a^coJ + aiyk} {a\-^b'^-^h'^-1bhy)i where the integral extends to the whole circuit. 440.] Suppose in particular C to be one pole of a small magnet or solenoid CC whose length is 2 1, and whose middle point is at Q and distant a from the centre of the conductor. Let LGGB = A. Then evidently a^=- a + l cos A, i = ^ sin A. Also the effect on C becomes, i£ al + b^+h'^ = A', ^J ^{{h-by)i^a^x3^a^h\ (l + _/ + _ —^ + ...} 15 hHH Sbh A^ ia-JjJe "•" "a^ '^Y a* + ...), 44I-] PHYSICAL APPLICATIONS. 255 since for the whole circuit /ey +1 = 0, f&xy'^ = 0. If we suppose the centre of the magnet fixed, the vector axis of the couple produced by the action of the current on C is IV. {i cos A + ^ sin ^)j-M- If A, &c. be now developed in powers of I, this at once becomes ■77^^^ sin A .C 6 a^ cos A ISa^^^cos^A 3P (a2 + /J2)f -^1 ~ a'^ + A^ + (a^ + A^)" '^T^ SlHin^A 15 A^Psin^A _ {a + lcosA)lcosA y 5 a^ cos A n | " a^ + A^ +T (a2+F)2~ ^^^^ y ~ a'-^-A'^ >V Putting —I for I and changing the sign of the whole to get that for pole C , we have for the vector axis of the complete couple 4TrA2;sinA.f ^ ^2(4a2_F)(4-5 sin^A) ) which is almost exactly proportional to sin A if la = A and I be small. On this depends a modification of the tangent galvanometer. (Bravais, Ann. de CAimie, xxxviii. 309.) 441.] As before, the effect of an indefinite solenoid on a^ is GAaa^^ Va-^y Now suppose a^ to be an element of a small plane circuit, 8 the vector of the centre of inertia of its area, the pole of the solenoid being origin. Let y = 8 + jO, then a^ = p. The whole effect is therefore _ CAaai f r{b + p)p' 2« 7 T{5+pf _ CAA^aa^ / 38^>. where A^ and e^ are, for the new circuit, what A and e were for the former. Let the new circuit also belong to an indefinite solenoid, and let 6o be the vector joining the poles of the two solenoids. Then the mutual effect is 256 QUATERNIONS. [442. 2xx^ J ^m "^ n' ) _ CAA-^aa^ \ Ub^ - 2wx, {n^^°^{n^' which is exactly the mutual effect of two magnetic poles. Two finite solenoids, therefore, act on each other exactly as two magnets, and the pole of an indefinite solenoid acts as a particle of free magnetism, 442.] The mutual attraction of two indefinitely small plane closed circuits, whose normals are e and e^, may evidently be deduced by TTh twice diflFerentiating the expression -f=j-^ for the mutual action of the poles of two indefinite solenoids, making db in one differentiation II f and in the other || e^. But it may also be calculated directly by a process which will give us in addition the couple impressed on one of the circuits by the other," supposing for simplicity the first to be circular. Let A and B be the centres of inertia of the areas of A and B, « and e^ vectors normal to their planes, o- any vector radius of B, AB = p. Then whole effect on </, by §§ 432, 435, 'Tifi + crf 5 r^'^ I r/3 'V+ Tifi + .f V ^7^i^'^Hl + l^)+ Tl3^ i^ + ^^J But between proper limits, frir'rtSdu =-A:ir,r]re€^, for generally fn'n 86<r = -k{ Fr,crSda- +7.7,7. QfT<T<j'). Hence, after a reduction or two, we find that the whole force exerted by A on the centre of inertia of the area of B 443-J PHYSICAL APPLICATIONS. 257 This, as already observe^ may be at once found by twice differ- entiating m;52' ^^ ^^® same way the vector moment, due to A, about the centre of inertia of £, These expressions for the whole force of one small magnet on the centre of inertia of another, and the couple about the latter, seem to be the simplest that can be given. It is easy to deduce from them the ordinary forms. For instance, the whole resultant couple on the second magnet oc ^ T0^ may easily be shewn to coincide with that given by Ellis {Camh. Math. Journal, iv. 95), though it seems to lose in simplicity and capability of interpretation by such modifications. 443.] The above formulae shew that the whole force exerted by one small magnet M, on the centre of inertia of another m, consists of four terms which are, in order, 1st. In the line joining the magnets, and proportional to the cosine of their mutual inclination. 2nd. In the same line, and proportional to five times the product of the cosines of their respective inclinations to this line. 3rd and 4th. Parallel to { ]■ and proportional to the cosine of the M .... inclination o/" { ^ to the joining line. All these forces are, in addition, inversely as the fourth power of the distance between the magnets. For the couples about the centre of inertia of m we have 1st. A couple whose axis is perpendicular to each magnet, and which is as the sine of their mutual inclination. .2nd. A couple whose axis is perpendicular to m and to the line joining the magnets, and whose moment is as three times the product of the sine of the inclination qfm, and the cosine of the inclination o^M, to the joining line. In addition these couples vary inversely as the third power of the distance between the magnets. 258 QUATEENIONS. [444. [These results afford a good example of what has been called the internal nature of the methods of quaternions, reducing, as they do at once, the forces and couples to others independent of any lines of reference, other than those necessarily belonging to the system under consideration. To shew their ready applicability, let us take a Theorem due to Gauss.] 444.] If two small magnets he at right angles to each other, the moment of rotation of the first is anproximately twice as great when the axis of the second passes through the centre of the first, as when the axis of the first passes through the centre of the second. In the first case e |{ y3 J.ej^ ; C 2 C" therefore moment = ^T(efi-3€€i) = ^yeej. In the second eil|/3±e; C therefore moment = -=j-^Tee-^. Hence the theorem. 445.] Again, we may easily reproduce the results of § 442, if for the two small circuits we suppose two small mag^nets perpendicular to their planes to be substituted. (3 is then the vector joining the middle points of these magnets, and by changing the tensors we may take 2e and 2ej^ as the vector lengths of the magnets. Hence evidently the mutual effect which is easily reducible to as before, if smaller terms be omitted. If we operate with V. e^ on the two first terms of the unreduced expression, and take the difference between this result and the same with the sign of e^ changed, we have the whole vector axis of the couple on the magnet 2ei, which is therefore, as- before, seen to be proportional to 446.] We might apply the foregoing formulae with great ease to other cases treated by Ampere, De Montferrand, &c. — or to two finite circular conductors as in Weber's Dynamometer — but in general the only difficulty is in the integration, which even in some of the simplest cases involves elliptic functions, &c., &c. {Quarterly Math. Journal, 1860.) 448-] PHYSICAL APPLICATIONS. 259 447.] Let F{y) be the potential of any system upon a unit particle at the extremity of y. F{y) = c (1) is the equation of a level surface. Let the differential of (1) be Svdy=0, (2) then v is a vector normal to (1), and is therefore the direction of the force. But, passing to a proximate level surface, we have Svby = bC. ' Make by=xv, then —a;Tv^ = bC, Hence v expresses the force in magnitude also. (§ 368.) Now by § 435 we have for the vector force exerted by a small plane closed circuit on a particle of free magnetism the expression A , ZySye\ omitting the factors depending on the strength of the current and the strength of magnetism of the particle. Hence the potential, by (2) and (1), oc ' - oc Ty^ area of circuit projected perpendicular to y oc spherical opening subtended by circuit. The constant is omitted in the integration, as the potential must evidently vanish for infinite values of Ty. By means of Ampere's idea of breaking up a finite circuit into an indefinite number of indefinitely small ones, it is evident that the above result may be at once ex- tended to the case of such a finite closed circuit. 448.] Quaternions give a simple me- thod of deducing the well-known pro- perty of the Magnetic Curves. Let A, A be two equal magnetic poles, whose vector distance, 2 a, is bi- sected in 0, QQ' an indefinitely small magnet whose length is Ip , where p-= OP^ Then evidently, taking moments, S 2 260 QUATEENIONS. [449. r{p+a)p' _ r{p-a)p' T{p + af - ± T{p-aY' where the upper or lower sign is to be taken according as the poles are like or unlike. Operate by S. Vap, Sap{p + af—Sa{p + a)Sp'{p-ira) ^ , -.t, i ^^^ y(p^„)S = ± {s^°ie With -a], r or S.af {-^-^V{p-\-a)-= + {same with —a\, ^p + a^ i.e. SadU(p + a) = + SadU{p — a), Sa { U{p + a) + U{p—a)} = const., or cos Z OAP ± cos / OA'P = const., the property referred to. If the poles be unequal, one of the terms to the left must be multiplied by the ratio of their strengths. 4(49.] K the vector of any point be denoted by p = ix+Jt/ + iz, (1) there are many physically interesting and important transformations depending upon the effects of the quaternion operator „ ■ d . d , d ,. ^ = ^^+^^ + ^^^ (') on various functions of p. When the function of p is a scalar, the effect of V is to give the vector of most rapid increase. Its effect on a vector function is indicated briefly in § 369. 450.] We commence with one or two simple examples, which are not only interesting, but very useful in transformations. 7 * V/) = fiy- +&c.)(«« + &c.) =— 3, (3) ViTpf = n{TpY-'^VTp = n{TpY-^p; (5) and, of course, v^-^ = -^^^; (5)i Tp Tp^ ~ Tp"^ whence, V ^^ =- ^rj =- ^rr. (6) and, of course, V2y- = — VyY= (6)^ Also, Vp =-3 = TpVUp + VTp.Up = TpVUp-l, ■'■ ^^P = -T^ (7) 453-J PHYSICAL APPLICATIONS. 261 451.] By the help of the above results, of which (6) is especially useful (though obvious on other grounds), and (4) and (7) very remarkable, we may easily find the effect of V upon more complex functions. Thus, VSap=-V{aic-{-kc.) = -a, (1) Vrap = — VFpa =—V{pa—Sap) = 3a— a= 2a (2) Hence ^ ^ap _ 2a ZpVap _ 2ap^ + 3pF'ap _ ap^ — 3pSap . . T^^Tp^~'T^~ Tf^ " Tp^ ^' Hence „ Vap p^ Sahp — ZSapSphp Sahp ZSapSpbp » '^"P / \ '^•8PV^= jp =_______ = _6_.(4) This is a very useful transformation in various physical applica- tions. By (6) it can be put in the sometimes more convenient form S.hpV^=hS.aVy~ (5) And it is worthy of remark that, as may easily be seen, —S may be put for V in the left-hand member of the equation. 452.] We have also 'f7r.0py=V{^Syp-pSPy + ySpp] =-yfi + 3S^y-l3y ^SjSy. (1) Hence, if <j) be any linear and vector function of the form (j)p = a + ^F.fipy + mp, (2) i.e. a self-conjugate function with a constant vector added, then V(f>p = 2S^y—3m = scalar (3) Hence, an integral of Vo- = scalar constant, is <t = (l>p (4) If the constant value of Vo- contain a vector part, there will be terms of the form Fep in the expression for a; which will then ex- press a- distortion accompanied by rotation. (§371.) Also, a solution of V^" = « (where q and a are quaternions) is q = SCp+Ffp + (Pp. It may be remarked also, as of considerable importance in phy- sical applications, that, by (1) and (2) of § 451, V{S+ir)ap = 0, but we cannot here enter into details on this point. 453.] It would be easy to give many more of these transforma- tions, which really present no difiiculty ; but it is sufficient to shew 262 QUATERNIONS. [454. the, ready applicability to physical questions of one or two of those already obtained ; a property of great importance, as extensions of mathematical physics are far more valuable than mere analytical or geometrical theorems. Thus, if (7 be the vector-displacement of that point of a homo- geneous elastic solid whose vector is p, we have, j» being the con- sequent pressure producedj Vj9-)-W = 0, (1) whence <S'SpV^<j-= —SbpVp = 8jb, a complete differential (2) Also, generally, p = kSVa, and if the solid be incompressible S^cT= (3) Thomson has shewn {Caml. and Bub. Math. Journal, ii. p. 62), that the forces produced by given distributions of matter, electricity, magnetism, or galvanic currents, can be represented at every point by displacements of such a solid producible by external forces. It may be useful to give his analysis, with some additions, in a qua- ternion form, to shew the insight gained by the simplicity of the present method. 454. j Thus, if Scrbp = 8 =,- , we may write each equal to -stpv^^. This gives (T = —Vyj^, J-P the vector-force exerted by one particle of matter or free electricity on another. This value of o- evidently satisfies (2) and (3). Again, if S.hpVa = 6 j—g , either is equal to -8.hpV^ by (4) of §451. Here a particular case is Fap which is the vector-force exerted by an element a of a current upon a particle of magnetism at p. (§ 436.) 455.] Also, by §451 (3), Vap _ ap^ — ZpSap 4'58.j PHYSICAL APPLICATIONS. 263 and we see by §§ 435, 436 that this is the vector-force exerted by a small plane current at the origin (its plane being perpendicular to a) upon a magnetic particle, or pole of a solenoid, at p. This expres- sion, being a pure vector, denotes aii elementary rotation caused by the distortion of the solid, and it is evident that the above value of (T satisfies the equations (2), (3), and the distortion is therefore pro- ducible by external forces. Thus the effect of an element of a current on a magnetic particle is expressed directly by the displace- ment, while that of a small closed current or magnet is represented by the vector-axis of the rotation caused by the displacement. 456.] Again, let ^5pVV=8^. It is evident that a- satisfies (2), and that the right-hand side of the above equation may be written Va,: o.opv Hence a particular case is -^.8pv0^ and this satisfies (3) also. Hence the corresponding displacement is producible by external forces, and Vo- is the rotation axis of the element at p, and is seen as before to represent the vector-force exerted on a particle of mag- netism at p by an element a of a current at the origin. 457.] It is interesting to observe that a particular value of o- in this case is ^ (T — —\VSaUp—yjr' as may easily be proved by substitution. Again, if Sbpa- = — 8 ~^ > we have evidently o- = V -jfj • Now, as yj^ is the potential of a. small magnet a, at the origin, on a particle of free magnetism at p, o- is the resultant magnetic force, and represents also a possible distortion of the elastic solid by external forces, since Vo- = V^o- = 0, and thus (2) and (3) are both satisfied. 458.] We conclude with some examples of quaternion integra- tion of the kinds specially required for many important physical problems. 264 QUATERNIONS. ^ [459. It may perhaps be useful to commence with a different form of definition of the operator V, as we shall thus, if we desire it, entirely avoid the use of ordinary Cartesian coordinates. For this purpose we write where a is any unit-vector, the meaning of the right-hand opei'ator (neglecting its sign) being the rate of change of the function to which it is applied per unit of length in the direction of the unit- vector a. If a be not a unit-vector we may treat it as a vector-velocity, and then the right-hand operator means the- m^e of change per unit of time due to the change of position. . Let a, /3, y be any rectangular system of unit-vectors, then by a fundamental quaternion transformation V = — aSaV — ySiS/SV — ySyV = ad^ + ^d^ ^ ydy , which is identical with Hamilton's form so often given above. (Lectures, § 620.) 459.] This mode of viewing the subject enables us to see at once that the effect of applying V to any scalar function of the position of a point is to give its vector of most rapid increase. Hence, when it is applied to a potential u, we have Vu = vector-force at p. It u be a velocity-potential, we obtain -the velocity of the fluid element at p ; and if w be the temperature of a conducting solid we obtain the flux of heat. Finally, whatever series of surfaces is repre- sented by u = C, the vector Vu is the normal at the point p, and its length is inversely as the normal distance at that point between two consecutive sur- faces of the series. Hence it is evident that S.dpVu =—du, or, as it may be written, —S.dpV= d; the left-hand member therefore expresses total differentiation in virtue of any arbitrary, but small, displacement dp. 460.] To interpret the operator V.aV let us apply it to a poten- tial function u. Then we easily see that u may be taken under the vector sign, and the expression F{aV)u = Y.aSJu denotes the vector, couple due to the force at p about a point whose relative vector is o. 462.] PHYSICAL APPLICATIONS. 265 Again, if o- be any vector function of p, we have by ordinary quaternion operations r(aV).(r = S.arT7(T-\.a£Vc7 — VSa<T. The meaning of the third term (in which it is of course understood that V operates on n- alone) is obvious from what precedes. It remains that we explain the other terms. 461.] These involve the very important quantities (not operators such as the expressions we have been hitherto considering), S.V(T and V.V<t, which form the basis of our investigations. Let us look upon <t as the displacement, or as the velocity, of a point situated at p, and consider the group of points situated near to that at p, as the quan- tities to be interpreted have reference to the deformation of the group. 462.] Let T be the vector of one of the group relative to that situated at p. Then after a small interval of time t, the actual coordinates become p + i^c and p + r+t{(T—8{TV)a) by the definition of V in § 458. Hence, if be the linear and vector function representing the deformation of the group, we have ^r = T—tS{TV)<T. The farther solution is rendered veiy simple by the fact that we may assume t to be so small that its square and higher powers may be neglected. If <^' be the function conjugate to <^, we have ^'t = T—tVST<T. Hence <^r = i(<^ + (^')r + i(0 — c^')'' = t--[-s(tv)o-+ v-Sro-]— ^ r.Trv(T. The first three terms form a self-conjugate linear and vector func- tion of r, which we may denote for a moment by utt. Hence (j)T = ■^r—rf'.rVVa; or, omitting f^ as above. Hence the deformation may be decomposed into — (1) the pure strain ■ST, (2) the rotation t „ 2 Thus the vector-axis of rotation of the group is 266 QUATERNIONS. [463. If we were content to avail ourselves of the ordinary results of Cartesian investigations, we might at once have reached this con- clusion by noticing that v% dzJ •'\dz Ax' \dx df and remembering as in (§ 362) the formulae of Stokes and Helmholtz. 463.] In the same way, as SV<T=—— — — — — dx dy dz^ we recognise the cubical compression of the group of points considered. It would be easy to give this a more strictly quaternionic form by employing the definition of § 458. Butj working with quaternions, we ought to obtain all our results by their help alone ; so that we proceed to prove the above result by finding the volume of the ellipsoid into which an originally spherical group of points has been distorted in time t. For this purpose, we refer again to the equation of deformation and form the cubic in ^ according to Hamilton's exquisite process. We easily obtaiuj remembering that <^ is to be neglected*, (i = ^^-{% — tSV(i)<^^ + {^ — nSV<j)^—{\—tSV<T), or = (^-1)2(^—1 + i!5Vcr). The roots of this equation are the ratios of the diameters of the ellipsoid whose directions are unchanged to that of the sphere. Hence the volume is increased by the factor 1— i!5Vo-, from which the truth of the preceding statement is manifest. * Thus, in Hamilton's notation, X, ;*, v being any three non-coplanar vectors, and m, m, , «i2 ^^ coefficients of the cubic, — ttSXnv = S-ip'f^^'iup'v =8.(\-ty8K<T)(ii-tVSiia){v-tVSr<r) ^S.(\-tVSKa)(yiiv-tVii^8va + tVy'78ii<T) =8.\iiv-t[S.iivV8\<T + 8.v\'78n<r + 8.\ii'VSva'i = S.\iiv-t8. l\8.iivV + ii8.v\V + v8.\iiV'\ a ^8.\iiv-tS.\iJiv8Va. 'miS.\iiv=S.\(t>'ii<l)'v + 8.fi^'v<l>'k + 8^vip'\(p'fi =8.K (ynv-tVnV8vtr+ tVvV8na) + &c. =8.Kiiv-t8.KnV8va-t8.v\V8n<i-¥ko. = 38.\nv-2t8Va8.\iiv. —m^S .\nv = 8 .\ii<p'v + 8. iiv^'\ + 8.v\(j> n =8.\iJi.v—t8.K/iV8va + &c. = S8.\iiv—t8V<r8.\iiv 465.] PHYSICAL APPLICATIONS. 267 464.] As the process in. last section depends essentially on the use of a non-conjugate vector function, with which the reader is less likely to be acquainted than with the more usually employed forms, I add another investigation. Let ■BT = ^T = T—tS{TV)(r. Then t = (f - V = t:j + tS (in- V) a. Hence since if, before distortion, the group formed a sphere of radius 1, we have Tt = 1, the equation of the ellipsoid is T{'!!T + tS(:!!TV)<T)= 1, or ■!!r^ + 2iS-nTVS^a- = — 1. This may be written S.wx^ = S.w {nr + i VSi!7(T + tS (in- V) <t) — — I, where x is now self- conjugate. Hamilton has shewn that the reciprocal of the product of the squares of the semiaxes is — 'S'-XWX'^. whatever rectangular system of unit-vectors is denoted by i, j, h. Substituting the value of x, we have —8.{i^tVSi(T^t8{iV)a) (y + &c.) (/^-|-&c.) = —S.{i■^r tVSia + tS («V) a){i+2 tiSVa— iS{iV)a- tVSicr) ^ l+2tSVa. The ratio of volumes of the ellipsoid and sphere is therefore, as before, 1 , = 1 - fSVcr. VI + 2tSV(T 465.] In what follows we have constantly to deal with integrals extended over a closed surface, compared with others taken through the space enclosed by such a surface ; or with integrals over a limited surface, compared with others taken round its bounding curve. The notation employed is as foUows. If Q, per unit of length, of surface, or of volume, at the point p, Q being any qua- ternion, be the quantity to be summed, these sums will be denoted by f/qds and Jf/qds, when comparing integrals over a closed surface with others through the enclosed space ; and by f/qds and /QTdp, when comparing integrals over an unclosed surface with others round its boundary. No ambiguity is likely to arise from the double use of 268 QUATERNIONS. [466. for its meaning in any case will be obvious from the integral with which it is compared. 466.J We have just shewn thatj if a- be the vector displacement of a point originally situated at p = ix+jy + kz, then S.Va- expresses the increase of density of aggregation of the points of the system caused by the displacement. 467.] Suppose, now, space to be uniformly filled with points, and a closed surface S to be drawn, through which the points can freely move when displaced. Then it is clear that the increase of number of points within the space 2, caused by a displacement, may be obtained by either of two processes — by taking account of the increase of density at all points within 2, or by estimating the excess of those which pass inwards through the surface over those which pass outwards. These are the principles usually employed (for a mere element of volume) in forming the so-called ' Equation of Continuity.' Let V be the normal to 2 at the point p, drawn outwards, then we have at once (by equating the two different expressions of the same quantity above explained) the equation ///S.Vads =//S.<rUvds, which is our fundamental equation so long as we deal with triple integrals. 468.] As a first and very simple example of its use, suppose o- to represent the vector force exerted upon a unit particle at p (of ordinary matter, electricity, or magnetism) by any distribution of attracting matter, electricityj or magnetism partly outsidcj partly inside 2. Then, if P be the potential at p, <r = VP, and if r be the density of the attracting matter, &c., at p, V(T=V^P = 4irr by Poisson's extension of Laplace's equation. Substituting in the fundamental equation, we have 4:i:///rds= 4:-nM=//S.VPUvds, where M denotes the whole quantity of matter, &c., inside 2. This is a well-known theorem. 469.] Let P and Pj be any scalar functions of p, we can of course find the distribution of matter, &c., requisite to make either of them 47I-J PHYSICAL APPLICATIONS. 269 the potential at p ; for, if fhe necessary densities be r and i\ re- spectively, we have as before Now V (P VPi) = VP VPj + P V^Pi , Hence, if in the above formula we put we obtain J/fS.VPVPJs = -///PV^P,ds+//PS.VP,Uvds, = -///P^^'Pds +//P,S.VP Uvds, which are the common forms of Greenes Theorem. Sir W. Thomson's extension of it follows at once from the same proof. 470.] If Pj be a many- valued function, but VPj single- valued, and if 2 be a multiply-connected* space, the above expressions require a modification which was first shewn to be necessary by Helmholtz, and first supplied by Thomson. For simplicity, suppose 2 to be doubly-connected (as a ring or endless rod, whether knotted or not). Then if it be cut through by a surface s, it will become simply-connected, but the surface-integrals have to be increased by terms depending upon the portions thus added to the whole surface. In the first form of Greenes Theorem, just given, the only term altered is the last : and it is obvious that if jo^ be the increase of P^ after a complete circuit of the ring, the portion to be added to the right-hand side of the equation is Pi/fS.VPUvds, taken over the cutting surface only. Similar modifications are easily seen to be produced by each additional complexity in the space 2. 471.] The immediate consequences of Green's theorem are well ' known, so that I take only one instance. Let P and P^ be the potentials of one and the same distribution of matter, and let none of it be within 2. Then we have ///{vpyds =f/ps.vpuvds, so that if VP is zero all over the surface of 2, it is zero all through the interior, i.e., the potential is constant inside 2. If P be the velocity-potential in the irrotational motion of an incompressible fluid, this equation shews that there can be no such motion of the * Called by Helmholtz, after Eiemann, mehrfach zusammenhdngend. In translating Helmholtz'a paper {Phil. Mag. 1867) I used the above as an English equivalent. Sir ■W.Thomson in his great paper on Yortex Motion {Trans. B. S-.E. 1868) uses the ex- pression "multiply-continuous." 270 QUATERNIONS. [472. fluid unlesB there is a normal motion at some part of the bounding surface, so long at least as 2 is simply-connected. Again, if 2 is an equipotential surface, f/f(ypfd, = Pf/s.vPUvds = Pf//v^Pds by the fundamental theorem. But there is by hypothesis no matter inside 2, so this shews that the potential is constant throughout the interior. Thus there can be no equipotential surface, not in- cluding some of the attracting matter, within which the potential can change. Thus it cannot have a maximum or minimum value at points unoccupied by matter. 472.] If, in the fundamental theorem, we suppose a- =Vt, which imposes the condition that S.V(T = 0, i.e., that the <r displacement is effected without condensation, it becomes //S.VrUvds =///S.V^Tds = 0. Suppose any closed curve to be traced on the surface 2, dividing it into two parts. This equation shews that the surface-integral is the same for both parts, the difference of sign being due to the fact that the normal is drawn in opposite directions on the two parts. Hence we see that, with the above limitation of the value of a, the double integral is the same for all surfaces bounded by a given closed curve. It must therefore be expressible by a single integral taken round the cui-ve. The value of this integral will presently be determined. 473.] The theorem of § 467 may be written ///V^Pds =//S.UvVPds =//S{UvV)Pds. From this we conclude at once that if ^ = iP+JP^ + kP^, (which may, of course, represent any vector whatever) we have ///V^ad,=//S{UvV)<Tds, or, if V^o- = T, ///rds=//S{U,V-')rds. This gives us the means of representing, by a surface-integral, a vector-integral taken through a definite space. We have already seen how to do the same for a scalar-integral — so that we can now express in this way, subject, however, to an ambiguity presently to be mentioned, the general integral 476.] PHYSICAL APPLICATIONS. 271 where q is any quaternion \^atever. It is evident that it is only in certain classes of cases that we can exnect a perfectly definite expression of such a volume-integral in terms of a surface-integral. 474.] In the above formula for a vector-integral there may present itself an ambiguity introduced by the inverse operation to which we must devote a few words. The assumption is tantamount to saying that, as the constituents of a- are the potentials of certain distributions of matter, &c., those of t are the corresponding densities each multiplied by 4 tt. If, therefore, r be given throughout the space enclosed by S, o- is given by this equation so far only as it depends upon the distribution within S, and must be completed by an arbitrary vector depending on three potentials of mutually independent distributions exterior to 2. But, if o- be given, t is perfectly definite ; and as Vo- = V-^Tj the value of V""^ is also completely defined. These remarks must be carefully attended to in using the theorem above : since they involve as particular cases of their application many curious theorems in Fluid Motion^ &c. 475.] As a particular case, the equation rV(r= of course gives V a- := u, a scalar. Now, if V be the potential of a distribution whose density is u, we have V'^v = 4Tr?<. We know that this equation gives one, and but one, definite value for V, so that there is no ambiguity in V = 4tV~^?<, and therefore o- = — - V« is also determinate. 47r 476.] This shews the nature of the arbitrary term which must be introduced into the solution of the equation rVcr=r. To solve this equation is (§ 462) to find the displacement of any one of a group of points when the consequent rotation is given. Here -SVr = -S. V FVo- = 5 W = ; so that, omitting the arbitrary term (§ 475), we have W=Vr, and each constituent of o- isj as above, determinate. 272 QUATERNIONS. [477. Thomson * has put the solution in a form which may be written if we understand by y*( ) dp integrating the term in da; as if y and z were constants, &c. Bearing this in mind, we have as verification, rv<r = i2ri[vTi+fr^dp^ = i{3T+/dpSVr}=T. 477.] We now come to relations between the results of integra- tion extended over a non-closed surface and round its boundary. Let IT be any vector function of the position of a point. The line-integral whose value we seek as a fundamental theorem is yS.adr, where t is the vector of any point in a small closed curve, drawn from, a point within it, and in its plane. Let o-Q be the value of a- at the origin of t, then a- = <rf,-S(TV)crQ, so that /S.o-dr =z/S.(a-o-SiTV)<io)dr. But fdr = 0, because the curve is closed ; and (Tait on Mectro-Di/namics, § 1 3, Quarterly Math. Journal, Jan. 1860) we have generally fS.TVS.Oadr = \S.V{TScr^T-<Tjr.TdT). Here the integrated part vanishes for a closed circuit, and \fT.TdT = dsUv, where ds is the area of the small closed curve, and Uv is a unit- vector perpendicular to its plane. Hence fS.cTf^dT = S.V(TgUv.ds. Now, any finite portion of a surface may be broken up into small elements such as we have just treated, and the sign only of the integral along each portion of a bounding curve is changed when we go round it in the opposite direction. Hence, just as Ampere did with electric currents, substituting for a finite closed circuit a network of an infinite number of infinitely small ones, in each contiguous pair of which the common boundary is described by equal currents in opposite directions, we have for a finite unclosed surface /S.adp = jyS.Vcrllv.ds. There is no diflSculty in extending this result to cases in which the * Electrostatics and Magnetism, § 521, or Phil. Trans., 1852. 478. J PHYSICAL APPLICATIONS. 273 bounding curve consists offletached ovals, or possesses multiple points. This theorem seems to have beeu first given by Stokes (Smith's Prize Esoam. 1854), in the form =//K'(|-f)-(£-£)+«(l-S))- It solves the problem suggested by the result of § 472 above. 478.] If a- represent the vector force acting on a particle of matter at p, —S.adp represents the work done while the particle is displaced along dp, so that the single integral /S.adp of last section, taken with a negative sign, represents the work done during a complete cycle. When this integral vanishes it is evident that, if the path be divided into any two parts, the work spent during the particle's motiou through one part is equal to that gained in the other. Hence the system of forces must be con- servative, i. e., must do the same amount of work for all paths having the same extremities. But the equivalent double integral must also vanish. Hence a conservative system is such that //dsS.V<TUv = 0, whatever be the form of the finite portion of surface of which ds is an element. Hence, as Vo- has a fixed value at each point of space, while Uv may be altered at will, we must have rvo- = 0, or Vo- = scalar. If we call X, T, Z the component forces parallel to rectangular axes, this extremely simple equation is equivalent to the well-known conditions dX_dY_Q ^_^^o ^_^=o Hy dx " ' dz dy ^ ' dx dz Returning to the quaternion form, as far less complex^ we see that Vo- = scalar = 4Trr, suppose, implies that o- = VP, where P is a scalar such that V2P= ^-nr; that is, P is the potential of a distribution of matter, magnetism, or statical electricity, of volume-density /. 274 QUATERNIONS. [479. Hence, for a non-closed path, under conservative forces -fS.a-dp = -fS.VPdp = -/S{dpV)P = /da,P=/dP = Pi-Po, depending solely on the values of P at the extremities of the path. 479.] A vector theorem, which is of great use, and which cor- responds to the Scalar theorem of § 473, may easily be obtained. Thus, with the notation already employed, /V.adr =/r{<T,-S(TV)<r,)dr, Now r{F.vr.TdT)(ra=-S{TS7)r.<TgdT-S{dTV)FT(ro, and d{S(,TV)r<ToT) = S{TV)r.<T^dT-\-S{dTV)ro^T. Subtracting, and omitting the term which is the same at both limits, we have fV, adr = — ¥.(¥. UvV) cr^ ds. Extended as above to any closed curve, this takes at once the form /r.(Tdp= -//ds r. ( r. Uvv) <t. Of course, in many cases of the attempted representation of a quaternion surface-integral by another taken round its bounding curve, we are met by ambiguities as in the case of the space- integral, § 474 : but their origin, both analytically and physically, is in general obvious. 480.] If P be any scalar function of p, we have (by the process of § 477, above) /Pdr =/{P,-S{rV)P,)dT = -/S.TVPo.dT. But r.W.rdT = drS.TV—TS.dTV, and dirSrV) = drS.TV + TS.drV. These give /Pdr = -^ {TSTV-F.FTdTV)Po = dsF.UvVPg, Hence, for a closed curve of any form, we have /Pdp=//dsr.Uvvp, from which the theorems of §§ 477, 479 may easily be deduced. 481.] Commencing afresh with the fundamental integral ///SV<rds=//S.aUvds, put a = UjS, and we have ///S^Vuds =//uS.^U'vds; 483-] PHYSICAL APPLICATIONS. 275 from which at once ///Vuas = f/uUvds, (1) or //fVTd,=/fUv.Tds (2) Putting WjT for r, and taking the scalar, we have f/f{SrVUi + u.^SVT)ds = f/n^Sr Uvds, whence ///(S{rV)(r + (T8.VT)ds = //(rSrUvds (3) 483.] As one example of the important results derived from these simple formulae, take the following, viz. : — ffr.{Y<j'Uv)Tds = /f<TSTUvds-//UvS<TTds, where by (3) and (1) we see that the right-hand member may be written = //f{8{rV)(7 + <rSVT-V S(TT)ds = -fffr.nv<j)Tds (4) This, and similar formulae^ are easily applied to find the potential and vector-force due to various distributions of magnetism. To shew how this is introduced, we briefly sketch the mode of expressing the potential of a distribution. 483.] Let or be the vector expressing the direction and intensity of magnetisation, per unit of volume, at the element ds. Then if the magnet be placed in a field of magnetic force whose potential is u, we have for its potential energy E = -ff/ScrVuds = ///uSV(rds-//tiS(rUvds. This shews at once that the magnetism may be resolved into a volume-density <S(V<7), and a surface-density —ScrUv. Hence, for a solenoidal distribution, S.'V(r = 0. What Thomson has called a lamellar distribution (PMl. Trans. 1852), obviously requires that Sadp be integrable without a factor ; i. e., that FVa- = 0. A complex lamellar distribution requires that the same expression be integrable by the aid of a factor. If this be u, we have at once FV[ua) = 0, or S.<tV(t=0. With these preliminaries we see at once that (4) may be written //F.{rcrUv)Tds=-///r.TFV(Tdi-///r.<TVTds+///Sav.Tds. Now, if T = V(-), where r is the distance between any external point and the element 276 QUATERNIONS. [484- ds, the last term on the right is the vector-force exerted by the magnet on a unit-pole placed at the point. The second term on the right vanishes by Laplace's equation, and the first vanishes as above if. the distribution of magnetism be lamellar, thus giving Thomson's result in the form of a surface integral. 484.] An application may be made of similar transformations to Ampere's Directrice de V action electrodynamique, which, § 432 above, is the vector-integral C^pdp P- where dp is an element of a closed circuit, and the integration extends round the circuit. This may be written -/ r.{dpv)l, so that its value as a surface integral is jjs {UvV)V-ds -JJuvV^ i ds. Of this the last term vanishes, unless the origin is in, or infinitely near to, the surface over which the double integration extends. The value of the first term is seen (by what precedes) to be the vector-force due to uniform normal magnetisation of the same surface. 2 485.] Also, since VUp = — -^ > we obtain at once whence, by difierentiation, or by putting p + a for p, and expanding in ascending powers of Ta (both of which tacitly assume that the origin is external to the space integrated through, i.e., that Tp nowhere vanishes), we have and this, again, involves 486.] The interpretation of these, and of more complex formulae of a similar kind, leads to many curious theorems in attraction and in potentials, Thus, from (1) of § 481, we have 488.] PHYSICAL APPLICATIONS, 277 which.^ves the attraction of a mass of density t in terms of the potentials of volume distributions and surface distributions. Putting this becomes 'TJv.<yds iim-iir-^=fp Tp JJJ Tp^ ~JJ Tp By putting cr = p, and taking the scalar, we recover a formula given above ; and by taking the vector we have r/fUvUpds = 0. This may be easily verified from the formula /Pdp = r//Uv.vPds, by remembering that VTp = Up. Again if, in the fundamental integral, we put (T = tUp, 487.] As another application, let us consider briefly the Stress- function in an elastic solid. At any point of a strained body let A. be the vector stress per unit of area perpendicular to i, n and v the same for planes per- pendicular to J and k respectively. Then, by considering an indefinitely small tetrahedron, we have for the stress per unit of area perpendicular to a unit-vector <a the expression kSia) + iJ.SJ(o + vSko> =-<j>a>, so that the stress across any plane is represented by a linear and vector function of the unit normal to the plane. But if we consider the equilibrium, as regards rotation, of an infinitely small parallelepiped whose edges are parallel to i, j, k respectively, we have (supposing there are no molecular couples) F{iK+JlJ. + kv) = 0, or 2 Fi^i = 0, or r.V^p = 0. This shews (§173) that in this case (j) is self-conjugate, or, in other words, involves not nine distinct constants but only six. 488.] Consider next the equilibrium, as regards translation, of any portion of the solid filling a simply-connected closed space. Let u be the potential of the external forces. Then the condition is obviously ff^ ( Vv) ds +fffdiVu = 0, where v is the normal vector of the element of surface ds. Here 278 QUATEENIONS. [489- the double integral extends over the whole boundary of the closed space, and the triple integral throughout the whole interior. To reduce this to a form to which the method of § 467 is directly applicable, operate by S.a where a is any constant vector whatever, and we have /y S .(paUvds + yy/ds SaVu = by taking advantage of the self-conjugateness of (p. This may be written ///ds{S.V<t>a + 8.dVv.) = 0, and, as the limits of integration may be any whatever, 8.V(t>a + S.aVu = (1) This is the required equation, the indeterminateness of a rendering it equivalent to tAree scalar conditions. There are various modes of expressing this without the a. Thus, if A be used for V when the constituents of <^ are considered, we may write Vu = -SVA.cjyp. In integrating this expression through a given space, we must remark that V and p are merely artificial symbols of construction, and therefore are not to be looked on as variables in the integral. 489.] As a verification, it may be well to shew that from this equation we can get the condition of equilibrium, as regards rotation, of a simply connected portion of the body, which can be written by inspection as //r.p<p{Uv)ds+///r.pVuds = 0. This is easily done as follows : (1) gives S.V<t>(r + S.crVu = 0, if, and only if, <r satisfy the condition S4{V)(T = 0. Now this condition is satisfied if cr = Kap where a is any constant vector. For S.<p{V)rap=-S.aF<j){V)p = S.aFV<t)p = 0. Hence ///'^s {S.Vcj) Fap + S.apVu) = 0, or f/dsS.apij}Uv+///ds S.apVu = 0. Multiplying by a, and adding the results obtained by making a in succession each of three rectangular vectors, we obtain the required equation. 490.] Suppose a- to be the displacement of a point originally at p, then the work done by the stress on any simply connected portion of the solid is obviously W=//S.(}>{Uv)<Tds, 49I-] PHYSICAL APPLICATIONS. 279 because <j) ( Uv) is the vector Toree overcome per unit of area on the element ds. This is easily transformed to W=///S.V(li<7ds. 491. J In this case obviously the strain-function is X (■nr) = ■ar — /S. (•srV)cr. Now if the strain be a mere rotation, in which case S.)(ZlT\T — S.-S7T = 0, whatever be the vectors ot and t, no work is done by the stress. Hence the expression for the work done by the stress must vanish if these conditions are fulfilled. Again, it is easily seen that when the strain is infinitely small the work must be a homogeneous function of the second degree of these critical quantities ; for, if it exist, it is essentially positive. Hence, even when finite, the work on unit-volume may be ex- pressed as » = 2.(5.x€X«'- -S""') {S-xrixn'-Sm'), where e, e', r), rf, which are in general functions of cr, become con- stant vectors if the stress is indefinitely small. When this is the case it is easy to see that, whatever be the number of terms under S, w involves twenty-one separate and independent constants only ; viz. the coefiicients of the homogeneous products of the second order of the six values of form S-XW-XJ—S'STT iovthe values i, J, ^ of ot or r. Supposing the strain to be indefinitely small, we have for the variation of to, the expression .+ ^{S.x^X^'-S,e'){S.bxr,xri'+S.bxr,'xv)- Now, by the first equation, we have SxOT = — *S'(t!rV)8(r. Hence, writing the result for one of the factors only, the variation of the whole work done by straining a mass is bJr= b///wds =/ffbw & = -^fffd,{8.xy\xr\-Sm) {-S.xe'5.(€V)6<r-f-S.xe^(€'V)8<7}. Now, if we have at the limits 8(7 = 0, i.e. if the surface of the mass is altered in a given way, we have obviously, fffdsS.'wS{€^)b<T = -///dsS.b(TS{iV)w. 280 QUATERNIONS [492- Hence Now any arbitrary change in o- will in general increase the amount of work done, so that we have = 2 [5(eV) {x«'('S.X'7X'?'-'Sw')} +'S(e'V) {xeC&XIX'?'-'^'/'?')}]. which is our equation for the determination of cr, as the constants e, i, t), rj' are dependent solely on the elastic properties of the sub- stance distorted, and may therefore be considered as known ; while X essentially involves o-. 492.] Since the algebraic operator when applied to any function of a;, simply changes x into x-\-U, it is obvious that if o- be a vector not acted on by „ . d . d , d dx •' dy dz we have ,-s.vy(p) =/(p + ^), whatever function /"may be. From this it is easy to deduce Taylor's theorem in one important quaternion form. If A bear to the constituents of o- the same relation as V bears to those of p, and if_/and F be any two functions which satisfy the commutative law in multiplication, this theorem takes the curious form ,-^^^f{p) F{a) =/(p + A) F{<t) = F{<t + V)f{p) ; of which a particular case is ,S^f(^,)F{y) =/{x + ±)FQ,) = F{y + ^)/W. The modifications which the general expression undergoes, when ,/and i''are not commutative, are easily seen. If one of these be an inverse function, such as, for instance, may occur in the solution of a linear differential equation, these theorems of course do not give the arbitrary part of the integral, but they often materially aid in the determination of the rest. Other theorems, involving operators such as e*^, e^-'W^ &e., &c. are easily deduced, and all have numerous applications. 493.] But there are among them results which appear startling from the excessively free use made of the separation of symbols. Of these one is quite sufficient to shew their general nature. Let P be any scalar function of p. It is required to find the difference between the value of P at p, and its mean value throughout 494-] PHYSICAL APPLICATIONS. 281 a very small sphere, of radius r and volume v, whicli has the ex- tremity of p as centre. From what is said above, it is easy to see that we have the fol- lowing expression for the required result : — where o- is the vector joining the centre of the sphere with the ele- ment of volume <?s, and the integration (which relates to o- and & alone) extends through the whole volume of the sphere. Expanding the exponential, we may write this expression in the form higher terms being omitted on account of the smallness of r, the limit of T<T. Now, symmetry shews at once that fff^rd, = 0. Also, whatever constant vector be denoted by a, ///{Sa^fds = -aV/f{S<rUafds. Since the integration extends throughout a sphere, it is obvious that the integral on the right is half of what we may call the moment of inertia of the volume about a diameter. Hence {8<TUafd^ = '"^^ ///<■ 5 If we now write V for a, as the integration does not refer to V, we have by the foregoing results (neglecting higher powers of r) l///(.-..v_i)p,,=_ilv^p, which is the expression given by Clerk-Maxwell*. Although, for simplicity, P has here been supposed a scalar, it is obvious that in the result above it may at once be written as a quaternion. 494.J If p be the vector of the element ds, where the surface density isfp, the potential at o- is f/dsfpFT{p-<r), F being the potential function, which may have any form whatever. By the preceding, § 492, this may be transformed into ffasfp,^-yFTp; ' London Math. Soc. Proc, vol. iii, no. 3^, 1871. 282 QUATERNIONS. [495- or, far more conveniently for the integration, into where A depends on the constituents of a in the same manner as V depends on those of p. A still farther simplification may be introduced by using a vector a-Q, which is finally to be made zero, along with its corresponding operator Aq, for the above expression then becomes where p appears in a comparatively manageable form. It is obvious that, so far, our formulae might be made applicable to any distribu- tion. We now restrict them to a superficial one. 495.] Integration of this last form can always be easily effected in the case of a surface of revolution, the origin being a point in the axis. For the expression, so far as the integration is concerned, can in that case be exhibited as a single integral dx<f)Xi'' p where (f> may be any scalar function, and x depends on the cosine of the inclination of p to the axis. And As the interpretation of the general results is a little troublesome, let us take the case of a spherical shell, the origin being the centre and the density unity, which, while simple, sufficiently illustrates the proposed mode of treating the subject. We easily see that in the above simple case, a being any constant vector whatever, and a being the radius of the sphere, /"+" 2 Tra J —a ■^<* Now, it appears that we are at liberty to treat A as a has just been treated. It is necessary, therefore, to find the effects of such opera- tors as TA, e"^'^, &c., which seem to be novel, upon a scalar function of To- ; or %, as we may for the present call it. %F' Now (rA)2i?'=-A2J = 2?"' + — , whence it is easy to guess at a particular form of TA. To be sure that it is the only one, assume 496.] PHYSICAL APPLICATIONS. 283 where </> and i|f are scalar functions of JC to be found. This gives = 4>^F" + (<^^' 4- v/'<#) + <i>i') F' + {<pf' + ^^) F. Comparing, we have 2 (^\/f' + i|'^ = 0. From the first, ^ = ± Ij whence the second gives '>/' = + — > the signs of ^ and \/f being alike. The third is satisfied identically. That is +yA = ^ + -- ~ a® St Also, an easy induction shews that ±(»)- = (a)"+5(»r Hence we have at once by the help of which we easily arrive at the well-known results. This we leave to the student*. 496.] As an elementary example of the use of V in connection with the Calculus of Variations, let us consider the expression A =/QTdp, where Tdp is an element of a finite are along which the integration extends, and Q is in general a scalar function of p and constants. We have bA z=/{bQTdp+QbTdp) =/{bQTdp- QS. Udpdbp) = -iqSUdphp-] +/{bQTdp+S.bpdiQUdp)), where the portion in square brackets refers to the limits only, and gives the terminal conditions. The remaining portion may easily be put in the form S/dp{d{QUdp)-VQ.Tdp). * Proc. B. S. E., 1871-2. 284 QUATERNIONS. [497- If the curve is to be determined by the condition that the varia- tion of A shall vanish, we must have, as 8p may have any direction, or, with the notation of Chap. IX, This simple equation shews that (1) The osculating plane of the sought curve contains the vector VQ. (2) The curvature at any point is inversely as Q, and directly as the component of V Q parallel to the radius of absolute curvature. 497.] As a first application, suppose A to represent the action of a particle moving freely under a systan of forces which have a potential*, so that Q := ^o, and p2 = 2 {P-H), where P is the potential, H the energy constant. These give TpVTp = QVQ = -VP, and qp'= p, so that the equation above becomes simply p + VP = 0, which is obviously true. 498.] If we look to the superior limit only, the first expression for 6^ becomes in the present case -{TpSUdptp'] = -Sphp. If we suppose a variation of the constant H, we get the following term from the unintegrated part thH. Hence we have at once Hamilton's equations of varying action in the forms y^ _ a and ^ = t. The first of these gives, by the help of the condition above, (VJ)2 = 2 {P-H), the well-known partial difierential equation of the first order and second degree. 499.] To shew that, if A be any solution whatever of this equa- tion, the vector VA represents the velocity in a free path capable of 502.] PHYSICAL APPLICATIONS. 285 being described under the acflon of the given system of forces, we -j^P = P =-VP=-\V{VAf = ~S{VA.V)VA. But ~'VA=-S{fiV)VA. A comparison shews at once that the equality VA = p is consistent with each of these vector equations. 500.] Again, if 5 refer to the constants only, J a(VJ)2 = S.VA1>VA =-lH by the differential equation. But we have also — - = t, which gives 17^-^) — — 'S'(pV)aJ = 'dH. These two expressions for 3 jy again agree in giving VA = p, and thus shew that the differential coefficients of A with regard to the two constants of integration must, themselves, be constants. We thus have the equations of two surfaces whose intersection determines the path. 501.] Let us suppose next that A represents the time of passagCj so that the brachistochrone is required. Here we have the other condition being as in § 497, and we have which may be reduced to the symmetrical form p+p-^VP/J = 0. It is very instructive to compare this equation with that of the free path as above, § 497. The application of Hamilton's method may be easily made, as in the preceding example. (Tait^ Trans. R. S. E., 1865.) 503.] As a particular case, let us suppose gravity to be the only force, then VP = a, a constant vector, so that 286 QUATERNIONS. [503- The form of this equation suggests the assumption where jo and q are scalars and Sap = 0. Substituting, we get -j)qseo^qt + {-P'>-p'^a^ian^qt) = 0, which gives joq = T^^ = p^T^a. Now let jo /3~^o = y ; this must be a unit-vector perpendicular to a and /3, so that ir^ = -^— , (cos at— yBm at), cosqt '• ^ • whence p = cos qt {cos qt + y sin qt)P~^ (which may be verified at once by multiplication). Finally, taking the origin so that the constant of integration may vanish, we have 2/3/8 = t+ — (siQ2g'^— ycos22'^), 2q which is obviously the equation of a cycloid referred to its vertex. The tangent at the vertex is parallel to /3j and the axis of symmetry to a. 503.] In the case of a chain hanging under the action of given forces Q = Pr, where P is the potential, r the mass of unit-length. Here we have also, of course, /Tdp = I, the length of the chain being given. It is easy to see that this leads, by the usual methods, to the equation -=- {{Pr + ii)p'} —rVP = 0, where u is a scalar multiplier. 504.J As a simple case, suppose the chain to be uniform. Then r may be merged in u. Suppose farther that gravity is the only force, then P = Sap, VP = —a, and -J- {{Sap+u)p'} +a = 0. Differentiating, and operating by Sp\ we find S.p'[p'{8ap'+^)+a'^ = 0; which shews that u is constant, and may therefore be allowed for by change of origin. 505.J PHYSICAL APPLICATIONS. 287 The curve lies obviously in% plane parallel to a, and its equation is {8apY + a^ s^ = const., which is a well-known form of the equation of the catenary. When the quantity Q of § 496 is a vector or a quaternion, we have simply an equation like that there given for each of the con- stituents. 505.] Suppose P and the constituents of a- to be functions which vanish at the bounding surface of a simply-connected space 2, or such at least that either P or the constituents vanish there, the others (or other) not becoming infinite. Then, by § 467, ///d,S.V{Pa) =//dsPSaUv = 0, if the integrals be taken through and over 2. Thus ///dsS.(rVP = -///dsPS.V<T. By the help of this expression- we may easily prove a very re- markable proposition of Thomson {Cam. and Dub. Maih. Journal^ Jan. 1848, or Reprint of Papers on Electrostatics, § 206.) To shew that there is one, and lut one, solution of the equation S.V{e^Vu)= 4ir>- where r vanishes at anminfinite distance, and e is any real scalar what- ever, continuous or discontinuous. Let V be the potential of a distribution of density r, so that V^v = 4 nr, and consider the integral q = —JjJ^s (eVu- -Vv) . That Q may be a minimum as depending on the value of u (which is obviously possible since it cannot be negative, and since it may have any positive value, however large, if only greater than this minimum) , we must have = ibQ =-///dsS.(e^Vu—Vv)Vbu = ///<^s bu S.V {e^Vu-Vv), by the lemma given above, =/y/dsbti {S.V {e^Vu)-4:T!r}. Thus any value of u which satisfies the given equation is such as to make Q a minimum. But there is only one value of w which makes Q a minimum ; for, let Qi be the value of Q when «j^ = w + (^ is substituted for this value of u, and we have 288 QUATERNIONS. [505- Qi = —JJJds. (eV (m + <^) - i V w) The middle term of this expression may, by the proposition at the beginning of this section, be written 2f//ds<^{SV{e^Vu)-4:T!r}, and therefore vanishes. The last term is essentially positive. Thus if % anywhere differ from u (except, of course, by a constant quan- tity) it cannot make Q a, minimum ; and therefore m is a unique solution MISCELLANEOUS EXAMPLES. 1. The expression Fo/3 Fyb + Fay Vh^ + TaS V^y denotes a vector. What vector ? 2. If two surfaces intersect along a common line of curvature, they meet at a constant angle. 3. By the help of the quaternion formulae of rotation, translate into a new form the solution (given in § 234) of the problem of inscribing in a sphere a closed polygon the directions of whose sides are given. 4. Express, in terms of the masses, and geocentric vectors of the sun and moon, the sun's vector disturbing force on the moon, and expand it to terms of the second order; pointing out the mag- nitudes and directions of the separate components. (Hamilton, Lectures, p. 615.) 5. J£ q = r^, shew that 2dq = 2dri = i {dr+Kqdrq-^)Sq-^ = i {dr + q-^drKq)Sq-'- = (drq + Kqdr)q-''{q + Kq)-^ = {drq + Kqdr){r+Tr)-^ _ dr+Uq-^drUq-^ _ drUq + Uq-^dr _ q-^{U'qdr + drUq-'^) ~ Tq{Uq+Uq-^) ~ q{Uq+Uq-^) " Uq+Uq-^ _ q-^{qdr + Trdrq~'^) _ drUq-{- Uq-^d^ _ drKq-^ +q-'^dr ~ Tq{Uq+Uq-^) ~ Tq{l + Ur) " iTUr 1 MISCELLANEOUS EXAMPLES. 289 2clq =^\clr+ r.Fdrjqlq-^ = j 3r -V.Vdrj q-'' \q- q q S^ q q S ^ = drq-^ + V. Vq-^ Vcl/r (l + -^ j-i) : and give geometrical interpretations of these varied expressions for the same quantity. {Ihid. p. 628.) 6. Shew that the equation of motion of a homogeneous solid of revolution about a point in its axis, which is not its centre of gravity, is BYp^-ASlp = Ypy, where 12 is a constant. {Trans. U. 8. E., 1869.) 7. Integrate the differfential equations : {a.) % + aq = h, where a and h are given quaternions, and and -v/f given linear and vector functions. (Tait, Proe. B.S.E., 1870-1.) 8. Derive (4) of § 92 directly from (3) of § 91. 9. Find the successive values of the continued fraction where i and j have their quaternion significations, and so has the values 1, 2, 3, &c. (Hamilton, Lectures, p. 645.) 10. If we have m. = f-A) c, where c is a given quaternion, find the successive values. For what values of c does u become constant ? {Ihid. p. 652.) 11. Prove that the moment of hydrostatic pressures on the faces of any polyhedron is zero, {a.) when the fluid pressure is the same throughout, {b.) when it is due to any set of forces which have a potential. 12. What vector is given, in terms of two known vectors, by the relation p-^ = \ {ar^ + yS'^) ? Shew that the origin lies on the circle which passes through the extremities of these three vectors. tr 290 QUATEENIONS. 13. Tait, Tram, and Proc. R.S.B., 1870-3. With the notation of §§ 467, 477, prove («•) ///S{aV)rds =//rSaUvds. (6.) I{ S{pV)T = -nT, (« + 3)///r& = -f/rSp Uvds. (e.) With the additional restriction V^r = 0, //S.mi2np+{n+3)p^V).Tds = 0. (d.) Express the value of the last integral over a non- closed surface by a line-integral. (e.) -/Tdp =f/ds8.UvV<T, if (7 = Udp all round the curve. {/.) For any portion of surface whose bounding edge lies wholly on a sphere with the origin as centre ffds8.{UpUvV).<r = 0, whatever be the vector o-. iff.) /rdpV.tr =//ds{UvV^-S{UvV)V)(T, whatever be o-. 14. Tait, Trans. B. S. U., 1873. Interpret the equation d(T = uqdpq~^, and shew that it leads to the following results V^cr = qVn q~^, V.Mj-i = 0, V^M* = 0. Hence shew that the only sets of surfaces which, together, cut space into cubes are planes and their electric images. 1 5. What problem has its conditions stated in the following six equations, from which ^, rj, ( are to be determined as scalar functions ot x, y, g, or oi p = is!+jy+kz'> V^i = 0, V^r, = 0, V^f = 0, SViVrj = 0, SVriVC= 0, SV^Vi = 0, „ . d . d , d where V = »^- + ?^- +/e-=- ■ dx '' dy dz Shew that they give the farther equations MISCELLANEOUS EXAMPLES. 291 Shew that (with a change OT origin) the general solution of these equations may be put in the form where <j(> is a self-conjugate linear and vector function, and £, rj, ( are to be found respectively from the three values of_/at any point by relations similar to those in Ex. 24 to Chapter IX. (See Lame, Journal de MatAematiqties, 1843.) 16. Shew that, if p be a planet's radius vector, the potential P of masses external to the solar system introduces into the equation of motion a term of the form S (pV)VP. Shew that this is a self-conjugate linear and vector function of p, and that it involves only Jive independent constants. Supposing the undisturbed motion to be circular, find the chief effects which this disturbance can produce. 17. In § 405 above, we have the equations ?a(OT + «^OT) = 0, 8a^ =0, d = aiFia, Ta = 1, where u>^ is neglected. Shew that with the assumptions bit Uif qz^i", a = qPq-'^, r = fi", •sr = qrrr-^q-^, we have /3 = 0, Tj3 = 1, S/3t=0, F0{T + n^T) = O, provided co*S«a— coj^ = 0. Hence deduce the behaviour of the Fou- cault pendulum without the x, y, and ^, jj transformations in the text. Apply analogous methods to the problems proposed at the end of § 401 of the text. 18. Hamilton, Bishop Law's Premmm Examination, 1862. [a.) If OABP be four points of space, whereof the three first are given, and not eoUinear ; if also oa = a, ob = /3, op = p ; and if, in the equation a a the characteristic of operation F be replaced by S, the locus of P is a plane. What plane ? {i.) In the same general equation, if F be replaced by V, the locus is an indefinite right line. What line ? (c.) If F be changed to K, the locus of p is a point. What point ? (d.) If F be made = TJ, the locus is an indefinite half-line, or ray. What raj^ ? 292 QUATERNIONS. (e.) If F be replaced by T, the locus is a sphere. What sphere ? {/.) If F be changed to TV, the locus is a cylinder of revo- lution. What cylinder ? {g.) If 2?' be made TVU, the locus is a cone of revolution. What cone ? [h.) If SU be substituted for F, the locus is one sheet of such a cone. Of what cone ? and which sheet ? («.) If i'' be changed to VU, the locus is a pair of rays. Which pair? 19. Hamilton, Bishop Law's Premium Examination, 1863. {a.) The equation Spp' + a^ — expresses that p and p' are the vectors of two points p and p', which are conjugate with respect to the sphere or of which one is on the polar plane of the other. (b.) Prove by quaternions that if the right line pp', connecting two such pointSj intersect the sphere, it is cut har- monically thereby. (c.) If p' be a given external point, the cone of tangents drawn from it is represented by the equation, irppy = a^p-py; and the orthogonal cone, concentric with the sphere, by i8ppy+a^p"' = 0. {d.) Prove and interpret the equation, T{np-a) = T{p-na\ if Tp = Ta. {e.) Transform and interpret the equation of the ellipsoid, y(tp + p/() = K2_t2. {/.) The equation {k^-I^Y = {l^ + K^)Spp' + 2SLpKp' expresses that p and p' are values of conjugate points, with respect to the same ellipsoid. (ff.) The equation of the ellipsoid may also be thus written, S,;p = 1, if {k'^-l^)^v = {i.-kYp+2iSkp+ 2kSip. {h.) The last equation gives also. MISCELLANEOUS EXAMPLES. 293 {i.) With the same sigiltfication of v, the differential equations of the ellipsoid and its reciprocal become Svdp — 0, Spdu = 0. {j.) Eliminate p between the four scalar equations, Sap = a, Spp = b, Syp = c, Sep = e. 20. Hamilton, Bishop Law^s Premium Examination, 1864. {a.) Let Aj^B-^j A^,^^, ... A^B„ be any given system of posited right lines, the 2n points being all given; and let their vector sum, AB = Aj^B^+A^B^+.-.+A^B^, be a line which does not vanish. Then a point H, and a scalar A, can be determined, which shall satisfy the quaternion equation, HAj^.A^Bi+... +HA^.A^B^ = h.AB ; namely by assuming any origin 0, and writing, Qjj_ jr OA-AA + ■ ■ • + OAn-A„B„ AiB^+...+A„B„ A^B,+ ... (b.) For any assumed point C, let Qc = CA^.A^B^ + . . . + CA^.A^B,, ; , then this quaternion sum may be transformed as follows, Qc= Qh + CH.AB = {7i + GH).AB ; and therefore its tensor is Tqc = {fi'' + CH^f.lB, in which AB and CH denote lengths. (c.) The least value of this tensor TQc is obtained by placing the point. C at H; if then a quaternion be said to be a minimum when its tensor is such, we may write min. Qc = Qj=r= h.AB; so that this minimum of Qc is a vector. {d.) The equation TQc = c = any scalar constant > TQh expresses that the locus of the variable point C is a spheric surface, with its centre at the fixed point H, and with a radius r, or CH, such that ' r.AB = {TQc^-TQH^)i = (c^ - h\ AB^)^ ; 294 QUATERNIONS. so that H, as being thus the common centre of a series of concentric spheres, determined by the given system of right lines, may be said to be the Central Point, or simply the Centre, of that system. (e.) The equation TFQc = Cj = any scalar constant > TQh represents a right cylinder, of which the radius divided by AB, and of which the axis of revolution is the line, VQc = Qh = h.AB; wherefore this last right line, as being the common axis of a series of such right cylinders, may be called the Central Axis of the system. (/".) The equation SQc = ^2 = ^°y scalar constant represents a plane ; and all such planes are parallel to the Central Plane, of which the equation is {g.) Prove that the central axis intersects the central plane perpendicularly, in the central point of the system. • (Ji,.) When the n given vectors A^B-i, ... A„B„ are parallel, and are therefore proportional to n sealars, b^,...6„, the scalar A and the vector Qh vanish ; and the centre H is then determined by the equation bi.HAi+i2SA+--- + h-SA„= 0, or by the expression, where is again an arbitrary origin. 21. Hamilton, Bishop Law's Premium Examination, 1860. {a.) The normal at the end of the variable vector p, to the surface of revolution of the sixth dimension, which is represented by the equation (p2-a2)3 = 27a==(p-a)*, (a) or by the system of the two equations, p2_a2=3<2„2^ (p_„)2 = ^3^2^ ^^,^ MISCELLANEOUS EXAMPLES. 295 and the tangent to the meridian at that point, are respectively parallel to the two vectors, • v = 2{p-a)-tp, and T=2{l-2i){p-a) + i^p; so that they intersect the axis a, in points of which the vectors are, respectively, 2a 2{l — 2f)a 2-t' -^"^ {2-tf-2 (b.) If dp be in the same meridian plane as p, then t{l t){i-f)dp=3Tdi, and s''f* =^~* up 3 Under the same condition. (e.) '^^ = 1(1-^). dp 3^ ' (d.) The vector of the centre of curvature of the meridian, at the end of the vector p, is, therefore, /„<?Dx-^ 3 V 6a — (4 — Op '' = P-<^dp) =P-2Y^t= 2(1-0 • {e.) The eiqjressions in Example 38 give v^ = a?t^{\-tf, T^ = aH%l-f)^4:-f); 9 9 a^t hence (cr—pY = -t^P', and dp''' = -.dt"^; the radius of curvature of the meridian is, therefore. and the length of an element of arc of that curve is = Tdp=zTa{-^fdt. {/.) The same expressions give thus the auxiliary scalar t is confined between the limits and 4, and we may write t = 2 vers $, where 5 is a real angle, which varies continuously from to 2Tr ; the recent expression for the element of arc becomes, there- fore, ds=3TaJd0* and gives by integration s = 6Ta{e-sme), if the arc s be measured from the point, say F, for which p = a, and which is common to all the meridians ; and the total periphery of any one such curve is = 12Tr Ta. 296 QUATERNIONS. {ff.) The value of o- gives i{<r^-a^) = 3aH{i-t), 16{Fa<r)^ = -a*f{i-tf ; if, then, we set aside the axis of revolution o, which is crossed by all the normals to the surface (a), the surface of centres of curvature which is touched by all those normals is represented by the equation, 4 (0-2 -a2)3 + 27 a2(rao-)2 = (b) {h.) The point F is common to the two surfaces (a) and (b), and is a singular point on each of them, being a triple point on (a), and a double point on (b) ; there is also at it an infinitely sharp cusp on (b), which tends to coincide with the axis a, but a determined tangent plane to (a), which is perpendicular to that axis, and to that cusp ; and the point, say.?", of which the vector =— a, is another and an exactly similar cusp on (b), but does not belong to (a). (j.) Besides the three universally coincident intersections of the surface (a), with any transversal, drawn through its triple point F, in any given direction y9, there are always three other real intersections, of which indeed one coincides with F if the transversal be perpendicular to the axis, and for which the following is a general formula : p=Ta.[Ua+ {28U{a^)iYU^']. {j.) 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THE MISSING FRAGMENT OF THE LATIN TRANSLATION of the FOURTH BOOK OF EZRA, discovered, and edited with an Introduction and Notes, and a facsimile of the MS., by Robert L. Bensly, M.A. Sub-Librarian of the University Library, and Reader in Hebrew, Gonville and Caius College, Cambridge. Demy Quarto. Cloth, 10s. "Edited with true scholarly complete- added a new chapter to the Bible, and, start- ness." — WestTttinsier Review. ling as the statement may at first sight ap- " Wer sich je mit dem 4 Buche Esra pear, it is no exaggeration of the actual fact, eingehender beschaftigt hat, wtrd durch die if by the Bible we understand that of the obige, in jeder Beziehung musterhafte Pub- larger size which contains the Apocrypha, lication in freudiges Erstaunen versetzt wer- and if the Second Book of Esdras can be den." — Theologische Literaturzeiiung. fairly called a part of the Apocrypha."^ "It has been said of this book that it has Saturday Review. THEOLOGY-(ANCIENT). SAYINGS OF THE JEWISH FATHERS, comprising Pirqe Aboth and Pereq R. Meir in Hebrew and English, with Critical and Illustrative Notes, By CHARLES Taylor, M.A. Fellow and Divinity Lecturer of St John's College, Cambridge, ^nd Honorary Fellow of King's College, London. Demy 8vo. cloth. loj. " It is peculiarly incumbent on those who tion of the Talmud. In other words, it is look to Jerome or Origen for their theology or the first instance of that most valuable and exegesis to learn something of their Jewish neglected portion of Jewish literature being predecessors. The New Testament abounds treated in the same way as a Greek classic with sa^ngs which remarkably coincide with, in an ordinary critical edition. . . The T^- or closely resemble, those of the Jewish mudic books, which have been so strangely Fathers; and these latter probably would neglected, we foresee will be the most im» furnish more satisfactory and frequent illus- portant aids of the future for the proper un- trations of its text than the Old Testament." derstanding of the Bible. . . The Sayings of — Saturday Review. the Jewish Fathers may claim to be scholar- « TL < >*■ t i_ * i_ i_ » 3 1 ly") aiid, moreover* of a scholarship unusually X. _,TheMasseketh Aboth stands at the thorough and finished. It is greatly to be head of Hebrew non-canonical writings. It hoped that this instalment is an earnest of IS of ancient date claiming to contain the future work in the same direction; the Tal- dicta of teachers who flourished from b. c. 200 ^ud iS a mine that will take years to work to the same year of our era. The precise out."— Dublin University Magazine. time of Its compilation in its present form is, , , . of course, in doubt. Mr Taylor's' explana- "A careful and thorough edition which tory and illustrative commentary is very full <loes credit to English scholarship, of a short and satisfactory."— i"/ffejffl/or. treatise from the Mishna, containing a series '*Ifwe mistake not, this is the first pre- cise translation into the English language , _ accompanied by scholarly notes, of any per- — "Contemfiorary Review. of sentences or maxims ascribed mostly ,to Jewish teachers immediately preceding, or immediately following the Christian era. . . " London: Cambridge Warehouse, 17 Paternoster Row, THE CAMBRIDGE UNIVERSITY PRESS. ' 5 THEODORE OF MOPSUESTIA'S COMMENTARY ON THE MINOR EPISTLES OF S. PAUL. The Latin Version with the Greek Fragments^ edited from the MSS. with Notes and an Introduction, by H. B. Swete, B.D., Rector of Ashdon, Essex, and late Fellow of Gonville and' Caius College, Cambridge. In Two Volumes. Vol. I., containing the Introduction, with Photographs of the MSS., and the Commentary upon Galatiahs — Colossians, will be ready shortly. SANCTI IREN^I EPISCOPI LUGDUNENSIS libros quinque adversus Hsereses, versione Latina cum Codicibus Claromontano ac Arundeliano -denuo coUata, prsemissa de placitis Gnosticorum prolusione, fragmenta necnon Grsece, Syriace, Armeniace, commentatione perpetua et indicibus,variis edidit W. Wigan Harvey, S.T.B. CoUegii Regalis olim Socius. 2 Vols. Demy Oflavo. i8j. M. MINUCII FELICIS OCTAVIUS. The text newly revised from the original MS., with an English Com- mentary, Analysis, Introdudlion, and Copious Indices. Edited by H. A. HOLDEN, LL.D. Head Master of Ipswich School, late Fellow of Trinity College, Cambridge. Crown Odtavo. js. 6d. THEOPHILI EPISCOPI ANTIOCHENSIS LIBRI TRES AD AUTOLYCUM edidit, Prolegomenis Versione Notulis Indicibus instruxit GuLIELMUS GiLSON Humphry, S.T.B. CoUegii Sandliss. Trin. apud Cantabri- gienses quondam Socius. Post Oftavo. 5J. THEOPHYLACTI IN EVANGELIUM S. MATTH^I COMMENTARIUS, edited by W. G. Humphry, B.D. Prebendary of St Paul's, late Fellow of Trinity College. Demy Oflavp. 7 J. 6d. TERTULLIANUS DE CORONA MILITIS, DE SPECTACULIS, DE IDOLOLATRIA, with Analysis and English Notes, by George Currey, D.D. Preacher at the Charter House, late Fellow and Tutor of St John's College. Crown Oflavo, S^-^ • THEOLOGY-(ENGLISH). WORKS OF ISAAC BARROW, compared with the Original MSS., enlarged with Materials hitherto unpubUshed. A new Edition, by A. Napier, M.A. of Trinity College, Vicar of Holkham, Norfolk. 9 Vols. Demy 0<5lavo. £2; 3^- London: Cambridge Warehouse, 17 Paternoster Row. PUBLICATIONS OF TREATISE OF THE POPE'S SUPREMACY, and a Discourse concerning the Unity of the Church, by Isaac Barrow. Demy Odlavo. Ts. 6d. PEARSON'S EXPOSITION OF THE CREED, edited by Temple Chevallier, B.D. late Fellow and Tutor of St Catharine's College, Cambridge. Second Edition. Demy Odlavo. 7J. 6d. AN ANALYSIS OF THE EXPOSITION OF THE CREED written by the Right Rev. Father in God, John Pearson, D.D. late Lord Bishop of Chester. Compiled, with some additional matter occasionally interspersed, for the use of the Students of Bishop's College, Calcutta, by W. H. Mill, D.D. late Principal of Bishop's College, and Vice-President of the Asiatic Society of Calcutta ; since Chaplain to the most Reverend Archbishop Howley ; and Regius Professor of Hebrew in the University of Cambridge. Fourth English Edition. Demy Odlavo, cloth. 5j. WHEATLY ON THE COMMON PRAYER, edited by G. E. Corrie, D.D. Master of Jesus College, Examining Chaplain to the late Lord Bishop of Ely. Demy Odlavo. fs. 6d. CiESAR MORGAN'S INVESTIGATION OF THE TRINITY OF PLATO, and of Philo Judaeus, and of the effedls which an attachment to their writings had upon the principles and reasonings of the Fathers of the Christian Church. . Revised by H. A. Holden,.LL.D. Head Master of Ipswich School, late Fellow of Trinity College, Cambridge. Crown Odlavo. 4J. TWO FORMS OF PRAYER OF THE TIME OF QUEEN ELIZABETH. Now First Reprinted. Demy Odlavo. 6d. **Froin 'Collections and Notes' 1867 — of Occasional Forms of Prayer, but it had 1876, by W. Carew Hazlitt (p. 340), we leam been lost ^ight of for 200 years. * By the that — *A very remarkable volume, in the kindness of the present possessor of this original vellum cover, and containing 25 valuable volume, containing in all 25 distinct Forms of Prayer of the reign of Elizabeth, publications, I am enabled to reprint in the each with the autograph of Humphrey Dyson, - ■■ • . . « has lately fallen into the hands of mjr friend Mr H. Pyne, It is mentioned specially in the Preface to the Parker Society's volume each with the autograph of Humphrey Dyson, following pages the two Forms of Prayer has lately fallen into the hands of mjr friend supposed to have been lost." — Bxtriwtfrom Mr H. Pyne, It is mentioned specially in the Preface, London: Cambridge Warehouse, 17 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 7 SELECT DISCOURSES. by John Smith, late Fellow of Queens' College, Cambridge. Edited by H.,G. Williams, B.D. late Professor of Arabic. Royal 06lavo. yj. 6^/, " The ' Select Discourses ' "of John Smith, "It is necessary to vindicate the distinc- coUeeted and published from his papers after tion of these men, because history hitherto his death, are, in my opinion, much the most has hardly done justice to them. They have considerable work left to us by this Cambridge been forgotten amidst the more noisy parties School [the Cambridge Platonists]. They of their time, between whom they sought to have a right to a place in English literary ' mediate.... What they really did for the cause history."— Mr Matthew Arnold, in the of religious thought has, never been ade- Contemporary Review. quately appreciated. They worked with too " Of aU the products of the Cambridge little combination and consistency. But it is School, the '"Select Discourses' are perhaps impossible in any real^study of the age not to the highest, as they -are the most accessible recognise the significance of their labours, or and the most widely appreciated. ..and indeed to fail to see how much the higher movement no spiritually thoughtful mind can read them of the national mind was due to them, while unmoved. They carry us so directly into an others carried the religious and civil struggle atmosphere of divine philosophy, luminous forward to its sterner is.>iues." — Principal with the richest lights of meditative genius... Tulloch, Rational Theology in England He was one of those rare thinkers in whom zw tke x-jth Century. largeness of view, and depth, and wealth of "We may instance Mr Henry Griffin poetic and speculative insight, only served to Williams's, revised edition of Mr John Smith's evoke more fully the religious spirit, and * Select Discourses,' which have won Mr while he drew the mould of his thought from Matthew Arnold's admiration,, as an example ProCinuSjL he vivified the substance of it from of worthy work for an University Press to St Paul,'* undertake." — Times. THE HOMILIES, with Various Readings, and the Quotations from the Fathers given at length in the Original Languages. Edited by G. E. CORRIE, D.D. Master o^ Jesus College. Demy Oftavo. "js. dd. DE OBLIGATIONE CONSCIENTI^ PR^LEC- TIONES decern Oxonii in Schola Theologica habitas a ROBERTO Sanderson, SS. Theologis ibidem Professore Regio. With English Notes, including an abridged Translation, by W. Wheweli,, D.D. late Master of Trinity College. Demy Odlavo. Ts. dd. ARCHBISHOP USHER'S ANSWER TO A JESUIT, with other Trails on Popery. Edited by J. Scholefield, M.A. late Regius Professor of Greek in the University. Demy 0£lavo. "js. 6d. WILSON'S ILLUSTRATION OF THE METHOD of explaining the New Testament, by the early opinions of Jews and Christians concerning Christ. Edited by T. TURTON, D.D. late Lord Bishop of Ely. Demy Odlavo. ^s. LECTURES ON DIVINITY delivered in the University of Cambridge, by John Hey, D.D. Third Edition, revised by T. TuRTON, D.D. late Lord Bishop of Ely. 2 vols. Demy Ocftavo. i^s. London: Cambridge Warehouse, i-j Paternoster Row. PUBLICATIONS OF ARABIC AND SANSKRIT. POEMS OF BEHA ED DIN ZOHEIR OF EGYPT. With a Metrical Translation, Notes and Introduction, by E. H. Palmer, M.A., Barrister-at-Law of the Middle Temple, Lord Almoner's Professor of Arabic and Fellow of St John's College in the University of Cambridge, 3 vols. Crown Quarto. Vol. I. The Arabic Text. ioj. 6d. \ Cloth extra, 15J. Vol. II. English Translation, ioj. 6^/.; Cloth extra, 15 j. "Professor Palmer's activity in advancing Arabic scholarship has formerly shown itself in the production of his excellent Arabic Grammar, and his Descriptive Catalogue of Arabic MSS. in the Library of Trinity Col- lege, Cambridge. He has row produced an admirable text, which illustrates in a remark- able manner the flexibility and graces of the language he loves so well, and of which he seems to be perfect master.... The Syndicate of Cambridge University must not pass with- out the recognition of their liberality in bringing out, in a worthy form, so important an Arabic text. It is not the first time that Oriental scholarship has thus been wisely subsidised by Cambridge." — Indian Mail. "It is impossible to quote this edition with- out an expression of admiration for the per- fection to which Arabic typography has been brought in England in this magnificent Ori- ental work, the production of which redo,unds to the imperishable credit of the University of Cambridge. It may be pronounced one of the most beautiful Oriental books that have ever been printed in Europe : and the learning of the Editor worthily rivals the technical get-up of the creations of the soul of one of the most tasteful poets of Islftm, the study of which will contribute not a little to save the honour of the poetry of the Arabs. Here first we make the acquaintance of a poet who grives us something better than monotonous descriptions of camels and deserts, and may even be regarded as superior in charm to al Mutanabbi." — Mythologv among the He- brews {Engl. TransL), p. 194. ' "Professor Palmer has produced the com- plete works of Behi-ed-din Zoheir in Arabic, and has added a second volume, containing an .English verse translation of the whole. It is only fair to add that the book, by the taste of its arabesque binding, as well as by the beauty of the typography, which reflects great credit on the Cambridge Uni- versity Press, is entitled to a place in the drawing-room." — Times. "For ease and facility, for variety of metre, for imitation, either designed or un- conscious, of the style of several of our own poets, these versions deserve high praise We have no hesitation in saying that in both Prof. Palmer has made an addition' to Ori- ental literature for which scholars should be grateful ; and that, while his knowledge of Arabic is a sufficient guarantee for his mas- tery of the original, his English compositions are distinguished by versatility, command of language, rhythmical cadence, and, as we have remarked, by not unskilful imitations of the styles of several of our own favourite poets Hying and dtvid.."— Saturday Review. "This sumptuous edition of the poems of Behi-ed-din Zoheir is a very welcome addi- tion to the small series of Eastern poets' accessible to readers who are not Oriental- ists. ... In all there is that exquisite finish of which Arabic poetry is susceptible in so rare a degree. The form is almost always beau- tiful, be the thought what it may. But this, of course, can only be fully appreciated by Orientalists. And this brings us to the trans- lation. It. is excellently well done. Mr Palmer has tried to imitate the fall of the original in his selection of the English metre for the various pieces, and thus contrives to convey a faint idea of the graceful flow of the Arabic Altogether the inside of the book is worthy of the beautiful arabesque binding that rejoices the eye of the .lover of Arab axt."-— Academy. NALOPAKHYANAM, OR, THE TALE OF NALA j containing the Sanskrit Text in Roman Characters, followed by a Vocabulary in which each word is placed under its root, with references to derived words in Cognate Languages, and a sketch of Sanskrit Grammar. By the Rev. Thomas Jarrett, M.A. Trinity College, Regius Professor of Hebrew, late Professor of Arabic, and formerly Fellow of St Catharine's College, Cambridge. Demy Ofl;a,vo. loj. London: Cambridge Warehouse, ij Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 9 GREEK AND LATIN CLASSICS, &c. (See also pp. 20-23.) THE AGAMEMNON OF AESCHYLUS. With a Translation in English Rhythm, and Notes Critical and Ex- planatory, By Benjamin Hall Kennedy, D.D., Regius Professor of Greek. Crown OctavOj cloth. 6j. " One of the best editions of the master- tion of a great undertaking." — Sat. Rem, _ piece of Greek tragedy." — AthemEum. ' "Letme say thatlthinkitamostadmira- " By numberless other like happy and ble piece of the highest criticfsm. _. . . . Hikft weighty helps to a coherent and consistent your Preface extremely ; it is' just to the text and interpretation, Dr Kennedy has point." — Professor Paley. approved himself a guide to Aeschylus of " Professor Kennedy has conferred a boon certainly peerless calibre." — Contemp. Rev. on all teaphers of the Greek classics, by caus- "Itis needless to multiply proofs of the ing the substance of his lectures at Cam- value of this volume alike to the poetical bridge on the Agamemnon of ^schylus to translator, the critical scholar, and the ethical be pubIished...This edition of the Agamemnon' student. We must be contented to thank is one which no classical master should be Professor Kennedy for his admirable execu- without." — Examiner. HEPI AIKAIOSTNHS. THE FIFTH BOOK OF THE NICOMACHEAN' ETHICS OF ARISTOTLE. Edited by Henry Jackson, M.A., Fellow of Trinity College, Cambridge. Demy Octavo, cloth. 6j. "It is not too much to say that some of Scholars will hope that this is not the only the points he discusses have never had so portion of the Aristotelian writings which he much light thrown upon them, before. ... is likely to edit." — AikentEum. PRIVATE ORATIONS OF DEMOSTHENES, with Introductions and English Notes, by F. A. Paley, M.A. Editor of Aeschylus, etc. and J. E. Sandys, M.A. Fellow and Tutor of St John's College, and Public Orator in the University of Cambridge. Part I. Contra Phormionem, Lacritum, Pantaenetum, Boeotum de Nomine, Boeotum de Dote, Dionysodorum. Crown Odlavo, cloth. 6j. " Mr Paley*s scholarship is sound and literature which bears upon bis author, and accurate, his experience of editing wide, and the elucidation of matters of daily life, inthe if he is content to devote his learning and delineation of which Demosthenes is so rich, abilities to the production of such manuals obtains full justice at his hands We as these they will be received with gratitude hope this edition may lead the way to a more throughout the higher schools of the country. general study of these speeches in schools Mr ^dys is deeply read in the German than has hitherto been ^ossihle.— Academy. Part II.' Pro Phormione, Contra Stephanum I. II. ; Nicostratum, Cononem, Calliclem. Ts.ed. " To give even a brief sketch of these in tfie needful help which enables us to sneecbes r/'ro Phormione and Contra Ste- form a sound estimate of the rights of the iham<m-\ would be incompatible with our case . It is long smce we have come ■Umits though we can hardly conceive a task upon a work cvmcrag more pams, scho ar- ■ imire 'useful to the classical or professional ship, and varied research and illustration than «rholar than to make one for himself. Mr Sandys s contribution to the ' Private T f Is a great boon to those who set them- Orations of Demosthenes . —Sat. Rev selves to unravel the thread of arguments ". ..... the edition reflects credit on nro and con to have the aid of Mr Sandys's Cambridge scholarship, and ought to be ex- excellent running commentary . . . .and no tenslvely used. '-Athen,Bum. one can say that he is- ever deficient London: Cambridge Warehouse, 17 Paternoster Row. 10 PUBLICATIONS OF THE BACCHAE OF EURIPIDES, with Introduction, Critical Notes, and Archaeological Illustrations, by J. E. Sandys, M.A., Fellow and Tutor of St John's College, Cam- bridge, and Public Orator. {Nearly ready, PINDAR. OLYMPIAN 'AND PYTHIAN ODES. With Notes Explanatory and Critical, Introductions and Introductory Essays. Edited by C. A. M. Fennell, M.A., late Fellow of Jesus College. Crown Oc- tavo, cloth. 9J. "Mr Fennell deserves the thanks of all classical students for his careful and scholarly edition of the Olympian and Pythian odes. He brings to his task the necessary enthu- siasm for his author, great industry, a sound judgment, and, in particular, copious and minute learning in comparative philology. To his qualifications in this last respect every page bears witness." — Athenmum. "Considered simply as a contribution to the study and criticism of Pindar, Mr Fen- nell's edition is a work of great merit. Bui it has a wider interest, as exemplifying the change which has come over the methods and aims o( Cambridge scholarship within the last ten or twelve years. . . . The short introductions and arguments to the Odes, which for so discursive an author as Pindar are all but a necessity, are both careful and acute. . . Altogether, this edition is awelcome and wholesome sign of the vitality and de- velopment of Cambridge scholarship, and we are glad to see that it is to be continued." — Saturday Reziew. THE NEMEAN AND. ISTHMIAN ODES. [Preparing, PLATO'S PH^DO, literally translated, by the late E. M, COPE, Fellow of Trinity College, Cambridge. Demy 0(Slavo. 5j, ARISTOTLE. THE RHETORIC. With a Commentary by the late E. M. CoPE, Fellow of Trinity College, Cambridge, revised and edited for the Syndics of the University Press by J. E. Sandys, M.A., Fellow and Tutor of St John's College, Cambridge, and Public Orator. With a biographical jyiemoir by H. A. J. MUNRO, M.A. Three Volumes, Demy Odlavo. £i, i is. 6d. " This work is in many ways creditable to the University of Cambridge. The solid and extensive erudition of Mr Cope himself bears none the less speaking evidence to the value of the tradition which he continued, if it is not equally accompanied by those qualities of speculative originality and independent judg- ment which belong more to the individual writer than to his school. And while it must ever be regretted that a work so laborious should not have received the last touches of its author, the warmest admiration is due to Mr Sandys, for the manly, unselfish, and un- flinching spirit in which he has performed his most difficult and delicate task. If an English student wishes to have a full conception of what is contained in the RJietoric of Aris- totle, to Mr Cope's edition he must go."— Academy.. " Mr Sandys has performed his arduous duties with marked ability and admirable tact. ...Besides the revision of Mr Cope's material already referred to in his own words, Mr Sandys has thrown in many useful notes ; none more useful than those that bring the Commentary up to the latest scholarship by reference to important works that have ap- peared since Mr Cope's illness put a period to his 'labours. When the original Com- mentary stops abruptly three chapters be- fore the end of the third book, Mr Sandys carefully supplies 'the deficiency, following Mr Cope's general plan and the slightest available indications of his intended treat- ment. In Appendices he has reprinted from classical journals several articles, of Mr Cope's ; and, what is better, he has given the best of the late Mr Shilleto's 'Adversaria.* In every part of his work — revising, " supple- menting, and completing — he has done ex- ceedingly well." — Examiner. * ' A careful examination of the work shows that the high expectations of classical stu- dents will not be disappointed. Mr Cope's * wide and minute acquaintance with all the Aristotelian writings,' to which Mr Sandys justly bears testimony, his thorough know- ledge of the important contributions of mo- dern German scholars, his ripe and accurate scholarship, and above all, that sound judg- ment and never-failing good sense which are the crowning merit of our best English edi- tions of the Classics, all combine to make this one of the most valuable additions to the knowledge of Greek literature which we have had for many years," — Spectator. ^ "Von der Rhetorik isteineneue Ausgabe mit sehr ausfuhrlichem Commentar erschie- nen. Derselbe enthalt viel schatzbares ...» Der Herausgeber verdient fur seine muhe- volle Arbeit unseren lebhaften Dank."— Susemihl in Bursian's Jahresherichi. • London : Cambridge Warehouse^ 1 7 Paternoster Row, THE CAMBRIDGE UNIVERSITY PRESS. ii * , P. VERGILI MARONIS OPERA cum Prolegomenis et Commentario Critico pro Syndicis Preli Academici edidit Benjamin Hall Kennedy, S.T.P., Graecae Linguae Professor Regius. Extra Fcap. Oflavo, cloth. i,s. M. T. CICERONIS DE OFFICIIS LIBRI TRES, with Marginal Analysis, an English Commentary, and copious Indices, by H. A. HOLDEN, LL.D. Head Master of Ipswich School, late Fellow of Trinity College, Cambridge, Classical Examiner to the University of London. Third Edition. Revised and considerably enlarged. Crown 0(flavo. <)s. M. TULLII CICERONIS DE NATURA DEORUM Libri Tres, with Introduction and Commentary by Joseph, B. Mayor, M.A., Professor of Classical Literature at King's College, London, formerly Fellow and Tutor of St John's College, Cambridge, together with a new collation of several of the English MSS. by J. H. Swain- SON, M.A., formerly Fellow of Trinity College, Cambridge. \Nearly Ready. MATHEMATICS, PHYSICAL SCIENCE, &c. THE ELECTRICAL RESEARCHES OF THE HONOURABLE HENRY CAVENDISH, F.R.S. Written between 1771 and 1781, Edited from the original manuscripts in the possession of the, Duke of Devonshire, K. G.,. by J. Clerk Maxwell, F.R.S. Demy 8vo. cloth. i8j. "This work, which derives a melancholy endish's results with those of modern experi- interest from the lamented death of the editor menters. In some instances they describe following so closely upon its publication, is a experiments undertaken by the editor for the valuable addition to the history of electrical express purpose of throwing light on Caven- research. . . . The papers themselves are most dish's methods of inve^igation. Every de- carefuHy reproduced, with fac-similes of the partment of editorial duty appears to have author's sketches of experimental apparatus. been most conscientiously performed ; audit A series of notes by the editor are appended, must have been no small satisfaction to Prof, some of them devoted to mathematical dis- Maxwell to see this goodly volume completed cussions, and others to a comparison of Cav- beforehislife'sworkwasdone."— ^M^«iyz^w, A TREATISE ON NATURAL PHILOSOPHY. By Sir W. THOMSON, LL.D., D.C.L., F.R.S., Professor of Natural Philosophy in the University of Glasgow, and P. G. Tait, M.A., Professor of Natural Philosophy in the University of Edinburgh. Vol. I. Part I. i6j. " In this, the second edition, we notice a could form within the time at our disposal large amount of new matter, the importance would be utterly inadequate." — Nature, of which is such that any opinion which we ELEMENTS OF NATURAL PHILOSOPHY. By Professors Sir W. THOMSON and P. G. Tait. Part I. 8vo. cloth, Second Edition, gj. " This work is designed especially for the trigonometry. Tyros in Natural Philosophy use of schools and junior classes in the Uni- cannot be better directed than by being told versities, the mathematical methods being to give their diligent attention to an intel- limited almost without exception to those of ligent digestion of the contents of this cxcel- the most elementary geometry, algebra, and lent vnde mecum."—Inn. London: Cambridge Warehouse, 17 Paternoster Row. PUBLICATIONS OF A TREATISE ON THE THEORY OF DETER- MINANTS AND THEIR APPLICATIONS IN ANALYSIS AND GEOMETRY, by Robert Forsyth Scott, M.A., of St John's College, Cambridge. \In the Press, HYDRODYNAMICS, A Treatise on the Mathematical Theory of the Motion of Fluids, by Horace Lamb, M.A., formerly Fellow of Trinity College, Cambridge ; Professorof Mathematics in the University of Adelaide. DemySvo. I2j. THE ANALYTICAL THEORY OF HEAT, By Joseph Fourier. Translated, with Notes, by A. Freeman, M.A. Fellow of St John's College^ Cambridge. Demy Octavo. i6j. "Fourier's treatise is one of the very few matics who do not follo'w with freedom a scientific books which can never be rendered treatise in any language but their own. It, antiquated by the progress of science. It is is a model of mathematical reasoning applied not only the first and the greatest book on to physical phenomena, and is remarkable for the physical subject of the conduction of the ingenuity of the analytical process em- Heat, but in every Chapter new views are ployed -by the author." — Contemporary opened up into vast fields of mathematical Revieiv^ October, 1878. speculation. "There cannot be two opinions as to the "Whatever text- books may be written, value and importance of the ThiQrie de la f'ving, perhaps, more succinct proofs 0/ Chalenr. It nas been called *an exquisite ourier's different equations, Fourier him- mathepiatical poem,' not once but many times, self will in all time coming retain his unique 'independently, by mathematicians of different prerogative of being the guide of his reader schools. Many of the 'very greatest of mo- into regions inaccessible to meaner men, how- dem mathematicians regard it, justly, as the ever ejcpert." — Extract from letter of Pro- key which first opened to them the treasure- fessor Clerk Maxwell. house of mathematical physics. It is still the " It is time that Fourier's masterpiece, text-book of Heat Conduction, and there The Analytical Theory of Heat^ trans- seems little present prospect of its being lated by Mr Alex. Freeman, shpuld be in- superseded, though it is already more than troduccd to those English students of Mathe- half a century old." — Nature. MATHEMATICAL AND PHYSICAL PAPERS, By George Gabriel Stokes, M.A., D.C.L., LL.D., F.R.S., Fellow of Pembroke College, and Lucasian Professor of Mathematics in the University of Cambridge. Reprinted from the Original Journals and Transactions, withAdditionalNotesbytheAuthor. Vol.1. \Nearlyready, An elementary TREATISE GN QUATERNIONS, By P. G. Tait, M.A., Professor of Natural Philosophy in the Univer- sity of Edinburgh. Second Edition. Demy 8vo. 14J. COUNTERPOINT. A Practical Course of Study, by Professor G. A. Macfarren, M.A., Mus. Doc. Second Edition, revised. Demy. Quarto, cloth. 7^. bd. A CATALOGUE OF AUSTRALIAN FOSSILS (including Tasmania and the Island of Timor), Stratigraphically and Zoologically arranged, by Robert Etheridge, Jun., F.G.S., Acting Palaeontologist, H.M. Geol. Survey of Scotland, (formerly Assistant- Geologist, Geol. Survey of Victoria). Demy Odlavo, cloth, loj. dd. 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THE MATHEMATICAL WORKS OF ISAAC BARROW, D.D. , Edited by W. Whewell, D.D. Demy Octavo. ■ 7^^. &d. ASTRONOMICAL OBSERVATIONS made at the Observatory of Cambridge by the Rev. James Challis, M.A., F.R.S., F.R.A.S., Plumian Professor of Astronomy and Experi- mental Philosophy in the University of Cambridge, and Fellow of Trinity College. For various Years, from 1846 to i860. ASTRONOMICAL OBSERVATIONS from 1861 to 1865. Vol. XXI. Royal 4to, cloth. 15^. LAW. A SELECTION OF THE STATE TRIALS. By J. W. Willis-Bund, M.A., LL.B., Barrister-at-Law, Professor of Constitutional Law and History, University College, London. Vol. I. Trials for Treason (1327 — 1660). Crown 8vo. cloth, i8j. THE FRAGMENTS OF THE PERPETUAL EDICT OF SALVIUS JULIANUS, collected, arranged, and annotated by Bryan Walker, M.A. LL.D., Law Lecturer ot St John's College, and late Fellow of Corpus. Christi College, Cambridge., Crown 8vo., Cloth, Price 6j. "This is one of the latest, we believe mentaries and the Institutes . . . Hitherto quite the latest, of the contributions made to the Edict has been almos,t inaccessible to legal scholarship by that revived study of the ordinary English student, and such a the Roman Law at Cambridge which is now student will be interested as well as perhaps so marked a feature in the industrial life surprised to find how abundantly the extant of the University. ■. . . In the present book fragments illustrate and clear up points which we have the fruits of the same kind of have.attracted his attention in the Commen- thorough and well-ordered study which was taries, or the Institutes, or the Digest."— brought to bear upon the notes to the Com- Law Times. London : Cambridge Warehouse, 1 7 Paternoster Row. 14 PUBLICATIONS OF THE COMMENTARIES OF GAIUS ANC RULES - OF ULPIAN. (New Edition, revised and enl^-rged.) With a Translation and Notes, by J. T.'Abdy, LL.D., Judge of County Courts, late Regius Professor of Laws in the University of Cambridge and Bryan Walker, M.A., LL.D., Law Lecturer of St John's College, Cambridge, formerly Law Student of Trinity Hall and ■Chancellor's Medallist for Legal Studies. Crown Odlavo, i6j. "As scholars and as editors Messrs Abdy "The number of books on various subjects and Walker have done their work well. of the civil law, which have lately issued from ..... For one thing the editors deserve the Press, shews that the revival of the study special commendation. They have presented of Roman Jurisprudence in this country is Gaius to the reader with few notes and those ^ genuine and increasing. The present edition merely by way of reference or necessary of Gaius and Ulpian from the Cambridge explanation. Thus the Roman jurist is University Press indicates that the Universi- allowed to speak for himself, and the reader ties are alive to the importance of the move- fecls that he is really studying Roman law ment."— Zaw youmaL in the original, and not a fanciful representa-- tion of it." — Athenesutn. THE INSTITUTES OF JUSTINIAN, translated with Notes by J. T. Abdy, LL.D., Judge of County Courts, late Regius Professor of Laws in the University of Cambridge, and formerly Fellow of Trinity Hall ; and Bryan Walker, M.A., LL.D., Law Lecturer of St John's College, Cambridge ; late Fellow and Lecturer of Corpus Christi College ; and formerly Law Student of Trinity Hall. Crown 0£lavo, i6j-, "We welcome here a valuable contribution attention is distracted from the subject-matter to the study of jurisprudence. The text of by the difficulty of struggling through the the /»j/zV»/f J- is occasionally perplexing, even language in which it is contained, it will be to practised scholars, whose knowledge of almost indispensable." — Spectator. classical models does not always avail them "The notes are learned and carefully com- in dealing with the technicalities of legal piled, and this edition will be found useful phraseology. Nor can the ordinary diction- to students." — La-w Times. aries be expected to furnish all the help that . **Dr Abdy and Dr Walker have produced is wanted. This translation will then be of a book which, is both elegant and useful."— great use. To the ordinary student, whose .Athenaeum. SELECTED TITLES FROM THE DIGEST, annotated by B. WALKER, M.A., LL.D. -Part I. Mandati vel Contra. Digest XVII. i. Crown 8vo., Cloth, 5^. "This small volume is published as an ex-' say that Mr Walker deserves credit for the periment The_ author proposes to publish an way in which he has performed the task un- aunotated edition and translation of several dertaken. The translation, as might be ex- books of the Digest if this one is received pected, is scholarly." Law Titnes, with favoiu:. We are. pleaded to be able to , Part II. Da Adquirendo rerum dominio and De Adquirenda velamit- tenda possessione. Digest XLI. i & 1 1. [Nearly Ready. GROTIUS DE JURE BELLI ET PACIS, with the Notes of Barbeyrac and others ; accompanied by an alsridged Translation of the Text, by W. Whewell, D.D. late Master of Trinity College. 3 Vols. Demy Odavo, I2j. The translation separate, 6s. London: Cambridge Warehouse, 17 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 15 HISTORY. LIFE AND TIMES OF STEIN, OR GERMANY AND PRUSSIA IN THE NAPOLEONIC AGE, by J. R. SeeLey, M.A., Regius Professor of Modern History in the University of Cambridge, with Portraits and Maps. 3 Vols. ' Demy 8vo. 48J. " If we could conceive anything similar to a protective system in the intellectual de- partment, we might perhaps look forward to a time when, our historians would raise the cry of protection for native industry. Of the unquestionably greatest German men of modern history — I speak of Frederick the Great,_ Goethe and Stein — the first two found long since in Carlyle and Lewes biographers who have undoubtedly driven their German competitors out of the field. And now in the year just past Professor Seeley of Cambridge has presented us with a biography of Stein whichj though it modestly declines competi- tion with German w6rks and disowns the presumption of teaching us Germans our own history, yet casts into the shade by its bril- liant superiority all that we have ourselves hitherto written about Stein..,. In five long chapters Seeley -expounds the legislative and administrative reforms, the emancipation of the person and the soil, the beginnings of free administration and free trade, in short the foundation of modern Prussia, with more exhaustive th.oroughness, with more pene- trating insight, than any one had done be- fore." — Deutsche Rundschau. ' " Dr Busch's volume has made people think and talk even more than usual of Prince Bismarck, and Professor Seeley's very learned work on Stein will turn attention to an earlier and an almost equally eminent German states- man It is soothing to the national self-respect to find a few Englishmen, such as the late Mr Lewes and Professor Seeley, doing for German as well as English readers what many German scholars have done for us." — Times. " In a notice of this kind scant justice can be done to a work like the one before us ; no short risumi can give even the most meagre notion of the contents of these volumes, which contain no page that is superfluous, and none that is uninteresting To under- stand the Germany of to-day one must study the Germany of many yesterdays, and now that study has been made easy by this work, to which no one can hesitate to assign a very high place among those recent histories which have aimed at original research." — Aihe- luBunt. "The book before us fills an important gap in English — nay, European — historical literature^ and bridges over the history of Prussia from the time of Frederick the Great to the days of Kaiser Wilhelm. It thus gives the reader standing ground, whence he may regard contemporary events in Germany in their proper historic light We con- gratulate Cambridge and her Professor of History on the appearance of such a note- worthy production. And we may add that it is something upon which we may congratulate England that on the especial field of the Ger- mans, history, on the history of their own country, by the use of their own literary weapons, an Englishman has produced a his- tory of Germany in the Napoleonic age far superior to any that exists in German." — Examiner. THE UNIVERSITY THE EARLIEST INJUNCTIONS OF OF CAMBRIDGE FROM TIMES TO THE ROYAL IS3S, by James Bass Mullinger, M.A. Demy 8vo. cloth (734 pp.), 12^. the University during the troublous times of the Reformation and the Civil War." — Athe- ndum. "Mr' Mullinger's work is one of great learning and research, which can hardly fail to become a standard book of reference on the subject. . . , We can most strongly recom- mend this book to our readers." — Spectator, "We trust Mr Mullinger will yet continue his history and bring it down to our own day. " — A cadetny, "He has brought together a mass of in- structive details respecting the rise and pro- gress, not only of his own University, but of all the principal Universities of the Middle Ages We hope some day that he may continue his labours, and give us a history of HISTORY OF THE COLLEGE OF ST JOHN THE EVANGELIST, by Thomas Baker, B.D., Ejected Fellow. Edited by Joh;^ E. B. MayoRj M,A., Fellow of St John's. Two Vols. Demy 8vo. 24J. "To antiquaries the book will be a source of almost inexhaustible amusement, by his- . torians it will be found a work of considerable service on questions respecting our social progress in past times ; and the care and thoroughness with which Mr Mayor has dis- charged,his editorial functions are creditable to his learning and industry." — Athenceum, " The, work displays very wide reading, and it will be of great use to members of the college and of the university, and, perhaps, of stil greater use to students of English history, ecclesiastipal, political, social, literary and academical, who have hitherto had to be content with'Dyer.'"— ^crtrf^;^^. " It may be thought that the history of a college cannot beparticularlyattractive. The two volumes before us, however, have some- thing more than a mere special interest for those who have been in any way connected with St John's Colleges Cambridge; they contain much which will be read with pleasure by a far wider circle.., The index with which Mr Mayor has furnished this useful work leaves nothing to be desired." — Spectator. London: Cambridge Warehouse^ 17 Faternoster Row. i6 PUBLICATIONS OF HISTORY OF NEPAL, translated by MuNSHi Shew Shunker Singh and Pandit Shri GUNANAND ; edited with an Introductory Sketch of the Country and People by Dr D. Wright, late Residency Surgeon at Kathma-ndu, and with facsimiles of native drawings, and portraits of Sir JUNG Bahadur, the King of Nepal, &c. Super-royal 8vo. Price 21s, ' "The Cambridge University Press have done well in publishing this work. Such translations are valuable not only to the his- torian but also to the ethnologist; Dr Wright's Introduction is based on personal inquiry and observation, is written intelli- gently and candidly, and adds much to the value of the volume. The coloured litho- graphic plates are interesting." — Nature. The history has appeared at a very op- portune moment. . .The volume. . .is beautifully J>rinted, and supplied with portraits of Sit i^ung Bahadoor and others, and with excel- ent coloured sketches illustrating Nepaulese architecture and religion." — Examiner. " In pleasing contrast with the native his- tory are the five introductory chapters con- tributed by Dr "Wright himself, who saw as muchof Nepal during- his ten years' sojourn as the strict rules enforced against foreigners even by Jung Bahadur would let him see." — Indian Mail. "Von nicht geringem Werthe dagegen sind die Beigaben, welche Wright als 'Appendix' hinter der 'history* folgen iSsst, AufzSh- lungen namlich der in NepSl (iblichen Musik- Instrumente, Ackergerathe, Miinzen, Ge- wichte, Zeittheilung, sodann ein kurzes Vocabular in Parbatly^ und NewSri, einige New^rl songs mit Interlinear-Uebersetzung, eine KOnigsliste, und, last not least, ein Verzeichniss der von ihm mitgebrachten Sanskrit-Mss. , welche jetzt in der UniVersi- tats-Bibliothek in Cambridge deponirt sind." — A. Weber, LiteraturzeiUing, Jahrgang 1877, Nr. 26. *' On trouve le portrait et la g^n^alogie de Sir Jang Bahadur dans I'excellent ouvrage que vientde publier Mr Daniel Wright sous le titre de * History of Nepal, translated from the Parbatiya, etc.'"— M. Garcin db Tassv in La Langue et la Littirature Hin- doustanies in 1877. Paris, 1878. SCHOLAE ACADEMICAE: Some Account of the Studies at the English Universities in the Eighteenth Century. By Christopher Wordsworth, .M.A., Fellow of Peterhouse; Author of "Social Life at the English Universities in the Eighteenth Century." Demy octavo, cloth, 15J. "The general object of Mr Wordsworth's book is sufficiently apparent from its title. He has collected a great quantity of minute and curious information about the working of Cambridge institutions in the last century, with an occasional comparison of the corre- sponding state of things at Oxford. It is of course impossible that a book of this kind should be altogether entertaining as litera- ture. To a great extent it is iiurely a book of reference, and as such it will be of per- manent value for the historical knowledge of English education and learning." — Saturday Review. "In the work before us, which is strictly what it professes to be, an account of university stu- dies, we obtain authentic information upon the course and changes of philosophical thought in this country, upon the general estimation of letters, upon the relations of doctrine and science, upon the range and thoroughness of education, and we may add, upoft the cat- like tenacity of life of ancient forms.... The particulars Mr Wordsworth gives us in his exceilenli arrangement are most varied, in- teresting, and instructive. Among the mat- ters touched upon are Libraries, Lectures, the Tripos, the Trivium, the Senate House, the Schools, text-books, subjects of study, foreign opinions,^ interior life. We learn even of the various University periodicals that have had their day. 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In order to provide Text-books for School and Examination pur- poses, -the Cambridge University Press has arranged to publish the several books of the Bible in separate portions at a moderate price, with introductions and explanatory notes. The Very Reverend J. J. S. Perowne, D.D., Dean of Peter- borough, has undertaken the general editorial supervision of the work, and will be assisted by a staflf of eminent coadjutors. Some of the books have already been undertaken by the following gentlemen ; Rev. A. Carr, M.A., Assistant Master at Wellington College. Rev. T. K. Cheyne, Fellow ofBalliol College, Oxford. Rev. S. Cox, Nottingham. Rev. A. B. Davidson, D.D., Professor of Hebrew, Edinburgh. Rev. F. W. Farrar, D.D., Canon of Westminster. Rev. A. E. Humphreys, M.A., Fellolv of Trinity College, Cambridge. Rev. A. F. KiRKPATRiCK, M.A., Fellow of Tiinity College. Rev. J. J. Lias, M.A., Professor at St David's College, Lampeter. Rev. J. R. LuMBY, D.D., Norrisian Professor of Divinity. Rev. G. F. Maclear, D.D., Head Master of King's Coll. School, London. Rev. H. C. G. Moule, M.A., Fellow of Trinity College. Rev. W. F. Moulton, D.D., Head Master of the Leys School, Cambridge. Rev. E. H. Perowne, D.D., Master of Corpus Christi College, Cam- bridge, Examining Chaplain to the Bishop of St Asaph, The Ven. T. T. IPerowne, M.A., Archdeacon of Norwich. Rev. A. Plummer, M.A., Master of University College, Durham. Rev. E. H. Plumptre, D.D., Professor of Biblical Exegesis, King's College, London. Rev. W. Sanday, M.A., Principal of Bishop Hatfield Hall, Durham. Rev. W. SiMCOX, M.A., Rector of Weyhill, Hants. Rev. Robertson Smith, lA.A^, Professor of Hebrew, Aberdeen. Rev. A. W. Streane, M.A., Fellow of Corpus Christi Coll.,Cambridge. Rev. H.W.Watkins, M.A., Warden of St Augustine's Coll., Canterbury. Rev. G. H. Whitaker, M.A., Fellow of St John's College, Cambridge. Now Beady. Cloth, Extra Fcap. 8vo. THE BOOK OF JOSHUA. Edited by Rev. G. F. Maclear, D.D. • With 2 Maps. is. 6d. THE BOOK OF JONAH. By Archdn. Perowne. is. 6d. London: Cambridge Warehouse, 17, Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 19 THE CAMBRIDGE BIBLE FOR ^GTSaOiL^— Continued. THE GOSPEL ACCORDING TO ST MATTHEW. Edited by the Rev. A. Carr, M.'A. With 2 Maps. 2j. dd. THE GOSPEL ACCORDING TO ST MARK. Edited by the Rev. G. F. Maclear, D.D. (with 2 Maps), is. 6d. THE ACTS OF THE APOSTLES. By the Rev. Professor LuMBY, D.D. Parti. Chaps. I— XIV. With 3 Maps. IS. 6d. THE EPISTLE TO THE ROMANS. By the Rev. H. C. G. MouLE, M.A. y. 6d. THE FIRST EPISTLE TO THE .CORINTHIANS." By the Rev. Professor Lias, M.A. With a Map and Plan. 2s. THE SECOND EPISTLE TO THE CORINTHIANS. By the Rev. Professor Lias, M.A, is. THE GENERAL- EPISTLE OF ST JAMES. By the Rev. Professor Plumptre, D.D. is. 6d. THE' EPISTLES '^ OF ST PETER AND ST JUDE. By the Rev. Professor Plumptre, D.D. is. 6d. Preparing. THE GOSPEL ACCORDING TO ST LUKE. By the Rev. F. W. Farkar, D.D., late Fellow of. Trinity College, Cambridge, Canon of Westminster, and Chaplain in Ordinary to the Queen. , [Nearly ready. THE GOSPEL ACCORDING TO" ST JOHN. By the Rev. W. Sanday and the Rev. A. i?LUMMER, M.A. THE FIRST BOOK OF SAMUEL. By the Rev. A. F. KiRKPATRICK, M.A. THE BOOK OF JEREMIAH, ^y the Rev. A. W. Streane, M.A. {Nearly ready. THE BOOKS OF HAGGAI AND ZECHARIAH. By Archdeacon Perowne. In Preparation. THE CAMBRIDGE GREEK TESTAMENT, FOR SCHOOLS AND COLLEGES, with a Revised Text, based on the most recent critical authorities, and English Notes, prepared under the direction of the General Editor, THE VERY REVEREND J. J. S. PEROWNE, D.D., DEAN OF PETERBOROUGH. The books will be published separately, as in /-^^ " Cambridge Bible for Schools." London: Cambridge Warehouse, 17 Paternoster Row. PUBLICATIONS OF THE PITT PRESS SERIES. I. GREEK. THE ANABASIS OF XENOPHON, Book VI. With a Map and English Notes' by Alfred Pretor, M.A., Fellow of St Catharine's College, Cambridge ; Editor of Persius and Cicero ad Atticum Book I. Price 2S. bd. BOOKS I. III. IV. & V. By the same Editor, 2s. each. BOOK II. By the same Editor. Price 2s. 6d. "This little volume (III.) is on every account well suited, either for schools or for the Local Examinations."— Times. **Mr Pretbr*s * Anabasis of Xenophon, Book IV.' displays a^ union of accurate Cambridge scholarship, with experience of what is required by learners gained in examining middle-class schools. The text is large and clearly printed, and the notes explain all difficulties. . . . Mr Fretor's notes seem to be all that could be wished as regards grammar, geography, and other matters," — T/te Academy, AGESILAUS OF XENOPHON. The Text revised with Critical and Explanatory Notes, Introduction, Analysis, and Indices. By H. Hailstone, M.A., late Scholar of Peterhouse, Cambridge, Editor of Xenophon's Hellenics, etc. Cloth, is. 6d. ARISTOPHANES— RANAE. With English Notes and Introduction by W. C. Green, M.A., Assistant Master at Rugby School. Cloth. 3J. 6d. ARISTOPHANES— AVES. By the same Editor. New Edition. Cloth. 3J. 6d. '*The notes to both plays are excellent. Much has been done in these two volumes to render the study of Aristophanes a real treat to a boy instead of a drudgery, by helping him to under- stand the fun and to express it in his mother tongue.'* — The Examiner.. EURIPIDES. HERCULES FURENS. With Intro- ductions, Notes and Analysis. By J. T. Hutchinson, B.A., Christ's College, Cambridge, and A. Gray, B.A., Fellow of Jesus College, Cambridge. Cloth, extra fcap. 8vo. Price is. "Messrs Hutchinson and Gray have produced a careful and useful edition." — Saturday Review. LUCIANI SOMNIUM CHARON PISCATOR ET DE LUCTU with English Notes by W. E. Heitland, M.A., Fellow and Lecturer of St John's College, Cambridge, Editor of Cicero pro Murena, &c. New Edition, with Appehdix. y. dd. London: Cambridge Warehouse, 17 Paternoster Row. THE CAMBRIDGE UNIVERSITY PRESS. 21 II. LATIN. M. T. CICERONIS DE AMICITIA. Edited by J. S. Reid, M.L., Fellow of Gonville and Caius College, Cambridge, Price 3J, "MrReid has decidedly attained his aim, namely, *a thorough examination of the Latinlty of the dialogue.' . . . , ■ The revision of the text is most valuable, and comprehends sundry acute corrections. . , . "We do not think that the most careful search would yield us many oppor- tunities for carping. This volume^ like Mr Reid's other editions, is a solid gain to the scholarship of the country.— j4^Af;i(EKw, "A more distinct gain to scholarship is Mr Reid's able and thorough edition of the De Amiciiid of Cicero, a work of which, whether we regard the exhaustive introduction or the instructive and most suggestive commentary, it would be difficult to speak too highly The characters of the dialogue are happily and sufficiently sketched. When we come to the com- mentary, we are only amazed by its fulness in proportion to its bulk. Nothing is overlooked which can tend to enlarge the learner's general knowledge of Ciceronian Latin or to elucidate the text. We have not spac^ to examine the editor's few, but generally well founded, corrections of the texX."— Saturday Review. M. T. CICERONIS CATO MAJOR DE SENECTUTE. Edited by J. S. Reid, M.L. Price gj-. 6d. M. T. CICERONIS ORATIO PRO ARCHIA POETA. . Edited by J. S. Reid, M.L. Pi'ice u. 6d. " It is an admirable sfjecimen of careful editing. ^ An Introduction tells us everything we could wish to know about Archias, about Cicero's connexion with him, about the merits of the trial, and the genuineness of the speech. The text is well and carefully printed. The notes are clear and scholar-like. . . . No boy can master this little volume without feeling that he has advanced a long step in scholarship." — Tke Academy. "The best of them, to our raind, are Mr Reid's two volumes containing the Pro Arckid. FoetA SLtid. Pro Balbo q{ C\ce.to, The introductions, which deal with the circumstances of each speech, giving also an analysis of its contents and a criticism of its merits, are models of clear and concise statement, at once intelligible to junior students and useful for those who are more ad- vanced." — Guardian. M. T. CICERONIS PRO L. CORNELIO BALBO ORA- TIO. Edited by J. S. Reid, M.L. Fellow of Caius College, Cambridge. Price IS. 6d. "Mr Reid's Orations for Archias and for Balbus profess to keep in mind the training.of the student's eye for the finer and more delicate matters of scholarship no less than for the more obvious ; and not only deal with the commonplace notahilia of a Latin oration as they serve the needs of a commonplace student, but also point out the specialities of Cicero's subject-matter and modes of expression. . ^ We are bound to recognize the pains devoted in the annotation of these two orations to the minute and thorough study of their Latinity, both in the ordinary notes and in the textual is.-^^cj\.^'iCQ%'*— Saturday Re-uiew. _ . , , "Mr Reid's Pro Balbo is marked by the same qualities as his edition of the Pro Areata." — TAs Academy., QUINTUS CURTIUS. A Portion of .the History. (Alexander, in India.) By W. E. Heitland, M.A., Fellow and Lecturer of St John's College, Cambridge, and T. E. Raven, B.A.„ Assistant Master in Sherborne School. Price },s. 6d. "Equally commendabls as a genuine addition to the existing stock of school-books is Alexander in India, a compilation from the eighth and ninth books of Q. Curtms, edited for the Pitt Press by Messrs Heitland and Raven. . . . The work of Curtius has merits of its own which, in former generations, made it afavourite with English scholars, and which still make it a popular text-book in Continental schools The reputation of Mr Heitland is a sufficient guarantee for the scholarship of the notes, which are ample without being excessive, and the book is well furnished with all that is needful in the nature of maps, indexes, and ap- pendices." —Academy. London: Cambridge Warehouse, xi Paternoster Row. 22 PUBLICATIONS OF P. OVIDII NASONIS FASTORUM Liber VI. With a Plan of Rome and Notes by A. Sidgwick, M.A. Tutor of Corpus Christi College, Oxford. Price is. 6d, '• Mr Sidgwick's editing of the Sixth Hook of Ovid's Fasit furnishes a careful and serviceable volume for average students. 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